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cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 6 /plusminus put dup 7 /minusplus put dup 15 /bullet put dup 17 /equivalence put dup 20 /lessequal put dup 21 /greaterequal put dup 24 /similar put dup 26 /propersubset put dup 33 /arrowright put dup 41 /arrowdblright put dup 50 /element put dup 54 /negationslash put dup 80 /P put dup 91 /union put dup 98 /floorleft put dup 99 /floorright put dup 102 /braceleft put dup 103 /braceright put dup 106 /bar put dup 107 /bardbl put dup 110 /backslash put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMIB10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMIB10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 59 /comma put dup 83 /S put readonly def /FontBBox{-15 -250 1216 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBSY10 %!PS-AdobeFont-1.1: CMBSY10 1.00 %%CreationDate: 1992 Jul 23 21:21:18 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMBSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 50 /element put dup 54 /negationslash put readonly def /FontBBox{-27 -940 1332 825}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 45 /notbar put dup 78 /N put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 4 /floorleftbig put dup 5 /floorrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 40 /braceleftBigg put dup 48 /parenlefttp put dup 49 /parenrighttp put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 64 /parenleftbt put dup 65 /parenrightbt put dup 81 /producttext put dup 88 /summationdisplay put dup 89 /productdisplay put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI12 %!PS-AdobeFont-1.1: CMMI12 1.100 %%CreationDate: 1996 Jul 27 08:57:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 21 /lambda put dup 33 /omega put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 66 /B put dup 67 /C put dup 76 /L put dup 80 /P put dup 81 /Q put dup 83 /S put dup 84 /T put dup 96 /lscript put dup 97 /a put dup 98 /b put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put dup 117 /u put dup 118 /v put dup 120 /x put readonly def /FontBBox{-30 -250 1026 750}readonly def currentdict end 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 96 /lscript put dup 105 /i put dup 107 /k put dup 110 /n put dup 112 /p put dup 113 /q put readonly def /FontBBox{11 -250 1241 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR6 %!PS-AdobeFont-1.1: CMR6 1.0 %%CreationDate: 1991 Aug 20 16:39:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put readonly def /FontBBox{-20 -250 1193 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 73 /I put dup 76 /L put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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1991 Aug 18 21:06:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 19 /acute put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 50 /two put dup 51 /three put dup 58 /colon put dup 59 /semicolon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 86 /V put dup 87 /W put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 127 /dieresis put readonly def /FontBBox{-36 -251 1103 750}readonly def currentdict end currentfile eexec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B6758ACD94CC409925F5EC841EAF 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR12 %!PS-AdobeFont-1.1: CMR12 1.0 %%CreationDate: 1991 Aug 20 16:38:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 14 /ffi put dup 18 /grave put dup 19 /acute put dup 33 /exclam put dup 38 /ampersand put dup 39 /quoteright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 59 /semicolon put dup 61 /equal put dup 63 /question put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 91 /bracketleft put dup 93 /bracketright put dup 96 /quoteleft put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 125 /hungarumlaut put readonly def /FontBBox{-34 -251 988 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 97 /a put dup 98 /b put dup 99 /c put dup 105 /i put dup 115 /s put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB0B4E70F212EC976D65099D84E0D37A7A771C3101D6AD26A0513378F 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 1 /Delta put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 54 /six put dup 56 /eight put dup 61 /equal put dup 97 /a put dup 100 /d put dup 109 /m put dup 110 /n put dup 115 /s put dup 116 /t put dup 120 /x put readonly def /FontBBox{-36 -250 1070 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 21 /greaterequal put dup 48 /prime put dup 50 /element put dup 106 /bar put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI8 %!PS-AdobeFont-1.1: CMMI8 1.100 %%CreationDate: 1996 Jul 23 07:53:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 21 /lambda put dup 34 /epsilon put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 76 /L put dup 83 /S put dup 84 /T put dup 96 /lscript put dup 98 /b put dup 101 /e put dup 102 /f put dup 105 /i put dup 106 /j put dup 107 /k put dup 109 /m put dup 110 /n put dup 112 /p put dup 113 /q put dup 114 /r put readonly def /FontBBox{-24 -250 1110 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 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2D1E66EA65F152F9E4106AA7047026A850D8AA3A76354A220E4ED57E4E561691 7E59230789D9C4C708449F784AADA34A65C4A930AF51F5F6002E89D98C21315A 74 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 61 /slash put dup 110 /n put dup 117 /u put dup 118 /v put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 49 /one put dup 50 /two put dup 54 /six put dup 58 /colon put dup 61 /equal put dup 66 /B put dup 70 /F put dup 76 /L put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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end 703 2053 a 703 2117 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1990179) cvn H.B /ANN pdfmark end 703 2117 a Black Fo(])g(b)m(y)i(Ev)m(erest,)h(v)-5 b(an)32 b(der)h(P)m(o)s(orten,)g(Shparlinski,)i(and)d(W)-8 b(ard.)60 2291 y SDict begin H.S end 60 2291 a 60 2291 a SDict begin 14.5 H.A end 60 2291 a 60 2291 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 60 2291 a 157 x Fp(2)161 b(Preliminary)54 b(results)e(for)i Fg(S)8 b Fp(.)60 2668 y Fo(While)36 b(Theorem)p 0 .5 0 TeXcolorrgb 760 2668 a SDict begin H.S end 760 2668 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 809 2603 a SDict begin H.R end 809 2603 a 809 2668 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 809 2668 a Black Fo(\(b\))f(allo)m(ws)g(us)h(to)e(m)m(ultiply)i Fn(n)c Fm(2)f Fn(S)40 b Fo(b)m(y)c(the)f(primes)h(in)f Fm(P)3004 2683 y Fv(S;n)3148 2668 y Fo(to)f(sta)m(y)i(within)f Fn(S)6 b Fo(,)35 b(a)60 2788 y(vital)i(ingredien)m(t)h(in)e(pro)m(ving) h(Theorem)p 0 .5 0 TeXcolorrgb 1640 2788 a SDict begin H.S end 1640 2788 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 1689 2724 a SDict begin H.R end 1689 2724 a 1689 2788 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 1689 2788 a Black Fo(\(c\))g(is)g(to)f(b)s(e) g(able)h(to)f(do)h(the)f(opp)s(osite:)52 b(to)36 b(divide)i Fn(n)d Fm(2)f Fn(S)60 2908 y Fo(b)m(y)39 b(a)f(prime)h(and)f(sta)m(y)h (within)g Fn(S)6 b Fo(.)59 b(This)40 b(is)e(pro)m(vided)i(b)m(y)f(the)f (follo)m(wing)h(signi\014can)m(t)g(result,)i(due)d(to)60 3029 y(Somer,)d(generalising)g(sp)s(ecial)h(cases)f(due)g(to)f(Jarden)h ([)p 0 .5 0 TeXcolorrgb 2143 3029 a SDict begin H.S end 2143 3029 a 0 .5 0 TeXcolorrgb Fo(13)p 0 .5 0 TeXcolorrgb 2240 2964 a SDict begin H.R end 2240 2964 a 2240 3029 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0197383) cvn H.B /ANN pdfmark end 2240 3029 a Black Fo(,)g(Theorem)g(E],)g(Hoggatt)e (and)i(Bergum)f([)p 0 .5 0 TeXcolorrgb 3835 3029 a SDict begin H.S end 3835 3029 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 3933 2964 a SDict begin H.R end 3933 2964 a 3933 3029 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0349567) cvn H.B /ANN pdfmark end 3933 3029 a Black Fo(])60 3149 y(and)g(W)-8 b(alsh)34 b([)p 0 .5 0 TeXcolorrgb 572 3149 a SDict begin H.S end 572 3149 a 0 .5 0 TeXcolorrgb Fo(23)p 0 .5 0 TeXcolorrgb 670 3085 a SDict begin H.R end 670 3085 a 670 3149 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Wa) cvn H.B /ANN pdfmark end 670 3149 a Black Fo(])f(for)g(the)h(Fib)s(onacci)g(sequence)i(\(i.e.,)f Fn(P)43 b Fo(=)29 b(1,)34 b Fn(Q)29 b Fo(=)h Fm(\000)p Fo(1\))j(and)h(Andr)m(\023)-46 b(e-Jeannin)35 b([)p 0 .5 0 TeXcolorrgb 3734 3149 a SDict begin H.S end 3734 3149 a 0 .5 0 TeXcolorrgb Fo(2)p 0 .5 0 TeXcolorrgb 3783 3085 a SDict begin H.R end 3783 3085 a 3783 3149 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1131414) cvn H.B /ANN pdfmark end 3783 3149 a Black Fo(])e(for)60 3269 y(gcd\()p Fn(P)s(;)17 b(Q)p Fo(\))27 b(=)h(1.)60 3294 y SDict begin H.S end 60 3294 a 60 3294 a SDict begin 14.5 H.A end 60 3294 a 60 3294 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3) cvn H.B /DEST pdfmark end 60 3294 a Black 171 x Fk(Theorem)47 b(3)g Fo(\(Somer)40 b([)p 0 .5 0 TeXcolorrgb 1017 3465 a SDict begin H.S end 1017 3465 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 1115 3401 a SDict begin H.R end 1115 3401 a 1115 3465 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 1115 3465 a Black Fo(,)j(Theorem)e(5\(iv\)]\))p Fk(.)p Black 46 w Fl(L)-5 b(et)43 b Fn(n)e Fm(2)h Fn(S)6 b Fl(,)43 b Fn(n)f(>)e Fo(1)p Fl(,)k(with)e Fn(p)3078 3480 y Ft(max)3256 3465 y Fl(its)g(lar)-5 b(gest)42 b(prime)60 3585 y(factor.)i(Then,)33 b(exc)-5 b(ept)34 b(in)g(the)g(c)-5 b(ase)33 b(that)h Fn(P)48 b Fl(is)33 b(o)-5 b(dd)34 b(and)f Fn(n)i Fl(is)e(of)h(the)g (form)f Fo(2)2977 3549 y Fv(`)3030 3585 y Fm(\001)20 b Fo(3)34 b Fl(for)g(some)f Fn(`)27 b Fm(\025)i Fo(1)p Fl(,)k(we)60 3706 y(have)h Fn(n=p)440 3721 y Ft(max)603 3706 y Fm(2)28 b Fn(S)6 b Fl(.)206 3901 y Fo(W)-8 b(e)25 b(pro)s(duce)g(a)f(v)-5 b(arian)m(t)25 b(of)e(this)i(result)h(to)e(co)m (v)m(er)i(all)e(but)g(t)m(w)m(o)h(of)f(the)h(exceptional)h(cases,)i(as) c(follo)m(ws.)60 3921 y SDict begin H.S end 60 3921 a 60 3921 a SDict begin 14.5 H.A end 60 3921 a 60 3921 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.4) cvn H.B /DEST pdfmark end 60 3921 a Black 176 x Fk(Prop)s(osition)38 b(4.)p Black 41 w Fl(If)d Fn(P)48 b Fl(is)34 b(o)-5 b(dd)35 b(and)f Fn(n)28 b Fo(=)f(2)1719 4061 y Fv(`)1774 4097 y Fm(\001)22 b Fo(3)28 b Fm(2)g Fn(S)6 b Fl(,)34 b(wher)-5 b(e)35 b Fn(`)27 b Fm(\025)h Fo(3)p Fl(,)35 b(then)g Fn(n=)p Fo(2)27 b Fm(2)h Fn(S)6 b Fl(.)206 4293 y Fo(The)30 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4942 a Fo(lcm)1579 4958 y Fv(p)p Fu(j)p Fv(n)1698 4831 y Ff(\020)1757 4942 y Fn(p)1806 4901 y Fv(k)1845 4877 y Fh(0)1843 4917 y Fi(p)1883 4942 y Fn(`)1924 4957 y Fv(p)1964 4831 y Ff(\021)2051 4942 y Fm(j)27 b Fn(n)h Fo(=)2296 4847 y Ff(Y)2310 5063 y Fv(p)p Fu(j)p Fv(n)2440 4942 y Fn(p)2489 4901 y Fv(k)2526 4909 y Fi(p)2566 4942 y Fn(:)1242 b Fo(\(2\))p Black 1986 5289 a(3)p Black eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 60 -300 a Black Black 100 x Fo(But)31 b(since)g(for)f Fn(p)e(>)f 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Fm(\000)f Fn(\014)6 b Fo(\))1450 1493 y(=)27 b Fn(u)1609 1508 y Fv(p)1671 1493 y Fo(+)1778 1365 y Fv(p)p Fu(\000)p Ft(1)1769 1399 y Ff(X)1779 1609 y Fv(j)t Ft(=1)1929 1353 y Ff(\022)2002 1426 y Fn(p)2004 1562 y(j)2051 1353 y Ff(\023)2125 1493 y Fn(\013)2188 1452 y Fv(p)p Fu(\000)p Fv(j)2314 1493 y Fo(\()p Fm(\000)p Fn(\014)6 b Fo(\))2528 1452 y Fv(j)2565 1493 y Fn(=)p Fo(\()p Fn(\013)22 b Fm(\000)h Fn(\014)6 b Fo(\))1450 1857 y(=)27 b Fn(u)1609 1872 y Fv(p)1671 1857 y Fo(+)1769 1725 y Ft(\()p Fv(p)p Fu(\000)p Ft(1\))p Fv(=)p Ft(2)1822 1762 y Ff(X)1833 1972 y Fv(j)t Ft(=1)2036 1716 y Ff(\022)2110 1790 y Fn(p)2111 1925 y(j)2159 1716 y Ff(\023)2232 1857 y Fo(\()p Fm(\000)p Fo(1\))2434 1816 y Fv(j)2471 1857 y Fn(Q)2548 1816 y Fv(j)2584 1857 y Fn(u)2640 1872 y Fv(p)p Fu(\000)p Ft(2)p Fv(j)1450 2113 y Fm(\021)h Fn(u)1611 2128 y Fv(p)1750 2113 y Fo(\(mo)s(d)33 b Fn(p)p Fo(\))p Fn(;)60 2328 y Fo(giving)g(the)g(result.)p 3889 2328 4 66 v 3893 2266 59 4 v 3893 2328 V 3951 2328 4 66 v 206 2522 a(W)-8 b(e)33 b(ha)m(v)m(e)h(the)f(follo)m(wing.)206 2642 y(A)39 b(prime)h(is)g(called)g Fl(irr)-5 b(e)g(gular)39 b Fo(if)g(it)g(divides)i Fn(Q)e Fo(but)g(not)g Fn(P)14 b Fo(.)63 b(Clearly)40 b Fn(p)f Fe(-)f Fo(\001)h(for)g Fn(p)g Fo(irregular.)63 b(A)60 2763 y(prime)33 b(that)g(is)g(not)f (irregular)h(is)g(called)g Fl(r)-5 b(e)g(gular)p Fo(.)60 2782 y SDict begin H.S end 60 2782 a 60 2782 a SDict begin 14.5 H.A end 60 2782 a 60 2782 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6) cvn H.B /DEST pdfmark end 60 2782 a Black 180 x Fk(Lemma)43 b(6)e Fo(\(Lucas)c([)p 0 .5 0 TeXcolorrgb 905 2962 a SDict begin H.S end 905 2962 a 0 .5 0 TeXcolorrgb Fo(15)p 0 .5 0 TeXcolorrgb 1003 2898 a SDict begin H.R end 1003 2898 a 1003 2962 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1505161) cvn H.B /ANN pdfmark end 1003 2962 a Black Fo(,)g(pp.)f(295{297],)g(Carmic)m(hael)h([)p 0 .5 0 TeXcolorrgb 2213 2962 a SDict begin H.S end 2213 2962 a 0 .5 0 TeXcolorrgb Fo(7)p 0 .5 0 TeXcolorrgb 2262 2898 a SDict begin H.R end 2262 2898 a 2262 2962 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1502458) cvn H.B /ANN pdfmark end 2262 2962 a Black Fo(,)g(Theorem)g(XI)s(I],)g(Somer)f([)p 0 .5 0 TeXcolorrgb 3306 2962 a SDict begin H.S end 3306 2962 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 3404 2898 a SDict begin H.R end 3404 2898 a 3404 2962 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 3404 2962 a Black Fo(,)h(Prop)s(osition)60 3082 y(1\(viii\)],)k(Williams)f([)p 0 .5 0 TeXcolorrgb 855 3082 a SDict begin H.S end 855 3082 a 0 .5 0 TeXcolorrgb Fo(24)p 0 .5 0 TeXcolorrgb 953 3018 a SDict begin H.R end 953 3018 a 953 3082 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1632793) cvn H.B /ANN pdfmark end 953 3082 a Black Fo(,)g(pp.)g(83{84])e (\))p Fk(.)p Black 45 w Fl(If)i Fn(p)g Fl(is)h(an)f(o)-5 b(dd)40 b(prime)g(with)h Fn(p)e Fe(-)f Fn(Q)p Fl(,)k Fn(p)d Fe(-)f Fo(\001)p Fl(,)k(then)f Fn(p)d Fm(j)g Fn(u)3803 3097 y Fv(p)p Fu(\000)p Fv(")3930 3082 y Fl(,)60 3233 y(wher)-5 b(e)39 b Fn(")g Fl(is)g(the)g(L)-5 b(e)g(gendr)g(e)39 b(symb)-5 b(ol)1432 3123 y Ff(\020)1501 3194 y Ft(\001)p 1501 3210 59 4 v 1513 3268 a Fv(p)1570 3123 y Ff(\021)1630 3233 y Fl(.)58 b(On)39 b(the)g(other)h(hand,)f(if)h Fn(p)f Fl(is)g(irr)-5 b(e)g(gular)39 b(then)h(it)f(do)-5 b(es)39 b(not)60 3375 y(divide)34 b(any)h Fn(u)587 3390 y Fv(n)633 3375 y Fl(,)g Fn(n)28 b Fm(\025)g Fo(1)p Fl(.)206 3575 y Fo(The)34 b(follo)m(wing)f(result)g(follo)m(ws)h(easily)f(from)g (Lemmas)p 0 .5 0 TeXcolorrgb 2296 3575 a SDict begin H.S end 2296 3575 a 0 .5 0 TeXcolorrgb Fo(5)p 0 .5 0 TeXcolorrgb 2345 3511 a SDict begin H.R end 2345 3511 a 2345 3575 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5) cvn H.B /ANN pdfmark end 2345 3575 a Black 33 w Fo(and)p 0 .5 0 TeXcolorrgb 2567 3575 a SDict begin H.S end 2567 3575 a 0 .5 0 TeXcolorrgb Fo(6)p 0 .5 0 TeXcolorrgb 2616 3511 a SDict begin H.R end 2616 3511 a 2616 3575 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.6) cvn H.B /ANN pdfmark end 2616 3575 a Black Fo(.)60 3594 y SDict begin H.S end 60 3594 a 60 3594 a SDict begin 14.5 H.A end 60 3594 a 60 3594 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7) cvn H.B /DEST pdfmark end 60 3594 a Black 180 x Fk(Corollary)47 b(7.)p Black 45 w Fl(The)41 b(set)g Fm(P)1132 3789 y Ft(1st)1268 3774 y Fl(of)g(primes)g(that)g(divide)f(some)h Fn(u)2523 3789 y Fv(n)2569 3774 y Fl(,)i Fn(n)c Fm(\025)h Fo(1)h Fl(c)-5 b(onsists)41 b(pr)-5 b(e)g(cisely)40 b(of)h(the)60 3895 y(r)-5 b(e)g(gular)35 b(primes.)60 3993 y SDict begin H.S end 60 3993 a 60 3993 a SDict begin 14.5 H.A end 60 3993 a 60 3993 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8) cvn H.B /DEST pdfmark end 60 3993 a Black 101 x Fk(Lemma)k(8)f Fo(\(Somer)33 b([)p 0 .5 0 TeXcolorrgb 914 4094 a SDict begin H.S end 914 4094 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 1011 4030 a SDict begin H.R end 1011 4030 a 1011 4094 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 1011 4094 a Black Fo(,)g(Theorem)h(5\(ii\)]\))p Fk(.)p Black 41 w Fl(If)g Fn(m;)17 b(n)28 b Fm(2)h Fn(S)40 b Fl(then)35 b Fo(lcm)q(\()p Fn(m;)17 b(n)p Fo(\))27 b Fm(2)i Fn(S)6 b Fl(.)p Black 60 4294 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(Put)37 b Fn(`)c Fo(=)g(lcm)q(\()p Fn(m;)17 b(n)p Fo(\).)53 b(F)-8 b(rom)35 b(\()p 0 .5 0 TeXcolorrgb 1535 4294 a SDict begin H.S end 1535 4294 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 1584 4229 a SDict begin H.R end 1584 4229 a 1584 4294 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 1584 4294 a Black Fo(\))h(w)m(e)h(ha)m(v)m(e)g Fn(u)2089 4309 y Fv(n)2168 4294 y Fm(j)c Fn(u)2285 4309 y Fv(`)2318 4294 y Fo(,)j Fn(u)2437 4309 y Fv(m)2537 4294 y Fm(j)d Fn(u)2654 4309 y Fv(`)2686 4294 y Fo(,)k(so)f Fn(n)d Fm(j)g Fn(u)3081 4309 y Fv(n)3128 4294 y Fo(,)j Fn(m)e Fm(j)f Fn(u)3427 4309 y Fv(m)3529 4294 y Fo(and)j(hence)60 4414 y Fn(`)28 b Fm(j)f Fn(u)240 4429 y Fv(`)272 4414 y Fo(.)p 3889 4414 4 66 v 3893 4352 59 4 v 3893 4414 V 3951 4414 4 66 v 60 4512 a SDict begin H.S end 60 4512 a 60 4512 a SDict begin 14.5 H.A end 60 4512 a 60 4512 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark end 60 4512 a Black 102 x Fk(Lemma)39 b(9.)p Black 42 w Fl(We)c(have)p Black 146 4813 a(\(i\))p Black 48 w(If)f Fn(P)48 b Fl(is)35 b(o)-5 b(dd)34 b(and)h Fo(2)1040 4777 y Fv(`)1100 4813 y Fm(j)27 b Fn(u)1211 4828 y Ft(12)1286 4813 y Fl(,)34 b(wher)-5 b(e)35 b Fn(`)27 b Fm(\025)h Fo(1)p Fl(,)35 b(then)g Fo(2)2179 4777 y Fv(`)p Fu(\000)p Ft(1)2329 4813 y Fm(j)28 b Fn(u)2441 4828 y Ft(6)2480 4813 y Fl(;)p Black 116 5015 a(\(ii\))p Black 48 w(If)34 b Fo(3)28 b Fm(j)f Fn(u)594 5030 y Ft(8)p Fv(k)706 5015 y Fl(then)35 b Fo(3)27 b Fm(j)h Fn(u)1111 5030 y Ft(4)p Fv(k)1188 5015 y Fl(.)p Black 1986 5289 a Fo(4)p Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 60 -300 a Black Black Black 100 x Fl(Pr)-5 b(o)g(of.)p Black 48 w Fo(Using)34 b(the)f(notation)700 11 y Fn(P)777 -30 y Ft(\()p Fv(k)r Ft(\))902 11 y Fo(=)28 b Fn(P)14 b Fo(\()p Fn(\013)1184 -30 y Fv(k)1225 11 y Fn(;)j(\014)1330 -30 y Fv(k)1372 11 y Fo(\))28 b(=)f Fn(\013)1604 -30 y Fv(k)1669 11 y Fo(+)22 b Fn(\014)1828 -30 y Fv(k)1898 11 y Fo(=)27 b Fn(v)2048 26 y Fv(k)2091 11 y Fn(;)212 b(Q)2407 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TeXcolorrgb 1256 1655 a SDict begin H.R end 1256 1655 a 1256 1720 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 1256 1720 a Black Fo(\))f(and)f(2)c Fe(-)f Fn(u)1704 1735 y Ft(12)1811 1720 y Fo(for)32 b Fn(Q)h Fo(ev)m(en,)h(w)m(e)g(get)e(the)h(result.)p Black 125 1934 a(\(ii\))p Black 49 w(Since)j Fn(u)617 1949 y Ft(4)p Fv(k)725 1934 y Fo(=)c Fn(u)889 1883 y Ft(\(4\))889 1962 y Fv(k)983 1934 y Fn(u)1039 1949 y Ft(4)1077 1934 y Fo(,)k(it)e(is)i(enough)f(to)f(pro)m(v)m(e)i(that)f(if)f(3)d Fm(j)g Fn(u)2564 1883 y Ft(\(4\))2564 1962 y(2)p Fv(k)2693 1934 y Fo(and)k(3)c Fe(-)g Fn(u)3080 1949 y Ft(4)3153 1934 y Fo(then)36 b(3)31 b Fm(j)g Fn(u)3573 1883 y Ft(\(4\))3573 1962 y Fv(k)3666 1934 y Fo(.)50 b(No)m(w,)304 2055 y(w)m(orking)33 b(mo)s(dulo)g(3,)f Fn(P)1207 2019 y Ft(\(4\))1329 2055 y Fm(\021)c Fn(P)1511 2019 y Ft(2)1550 2055 y Fo(\(1)22 b Fm(\000)g Fn(Q)p Fo(\))h Fm(\000)f Fn(Q)2072 2019 y Ft(2)2112 2055 y Fo(,)33 b(using)g(\()p 0 .5 0 TeXcolorrgb 2465 2055 a SDict begin H.S end 2465 2055 a 0 .5 0 TeXcolorrgb Fo(3)p 0 .5 0 TeXcolorrgb 2514 1990 a SDict begin H.R end 2514 1990 a 2514 2055 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.3) cvn H.B /ANN pdfmark end 2514 2055 a Black Fo(\))f(and)h Fn(P)2851 2019 y Ft(4)2917 2055 y Fm(\021)28 b Fn(P)3099 2019 y Ft(2)3138 2055 y Fo(.)44 b(Th)m(us)1292 2352 y Ff(\022)1365 2425 y Fn(P)1442 2389 y Ft(\(4\))1365 2561 y Fn(Q)1442 2532 y Ft(\(4\))1537 2352 y Ff(\023)1638 2493 y Fo(=)1741 2199 y Ff(8)1741 2289 y(>)1741 2318 y(>)1741 2348 y(>)1741 2378 y(<)1741 2558 y(>)1741 2587 y(>)1741 2617 y(>)1741 2647 y(:)1830 2199 y(\000)1875 2236 y Ft(0)1875 2314 y(0)1911 2199 y Ff(\001)1956 2280 y Fn(;)50 b Fo(if)32 b Fn(P)41 b Fm(\021)28 b Fn(Q)g Fm(\021)g Fo(0;)1830 2344 y Ff(\000)1875 2380 y Ft(1)1875 2459 y(0)1911 2344 y Ff(\001)1956 2425 y Fn(;)50 b Fo(if)32 b Fn(P)41 b Fm(\021)28 b(\006)p Fo(1)p Fn(;)17 b(Q)28 b Fm(\021)g 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/View [/XYZ H.V] /Dest (theorem.10) cvn H.B /DEST pdfmark end 60 3435 a Black 123 x Fk(Prop)s(osition)40 b(10.)p Black 43 w Fl(If)d Fn(P)50 b Fl(is)37 b(o)-5 b(dd)37 b(and)f Fo(2)1601 3522 y Fv(`)1658 3558 y Fm(\001)23 b Fo(3)32 b Fm(2)g Fn(S)6 b Fl(,)38 b(wher)-5 b(e)36 b Fn(`)c Fm(\025)g Fo(3)p Fl(,)37 b(then)g Fo(2)2865 3522 y Fv(`)p Fu(\000)p Ft(1)3012 3558 y Fm(\001)24 b Fo(3)31 b Fm(2)h Fn(S)6 b Fl(.)52 b(In)36 b(p)-5 b(articular,)60 3679 y(then)35 b Fo(12)27 b Fm(2)h Fn(S)6 b Fl(.)p Black 60 3875 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(T)d(ak)m(e)37 b Fn(P)49 b Fo(o)s(dd.)k(Then)37 b Fn(P)1291 3839 y Ft(\(2\))1418 3875 y Fo(=)c Fn(P)1604 3839 y Ft(2)1667 3875 y Fm(\000)25 b Fo(2)p Fn(Q)36 b Fo(is)g(also)g(o)s(dd,)g(and)g(hence)h(so)f(are)g (all)f Fn(P)3425 3839 y Ft(\(2)3487 3815 y Fi(`)3518 3839 y Ft(\))3582 3875 y Fo(=)e Fn(v)3738 3894 y Ft(2)3773 3875 y Fi(`)44 b Fo(for)60 3995 y Fn(`)28 b Fm(\025)g Fo(0.)43 b(Then)34 b(for)e Fn(`)27 b Fm(\025)i Fo(3,)j(using)h(\()p 0 .5 0 TeXcolorrgb 1332 3995 a SDict begin H.S end 1332 3995 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 1381 3931 a SDict begin H.R end 1381 3931 a 1381 3995 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 1381 3995 a Black Fo(\))f(and)h Fn(u)1697 4010 y Ft(2)p Fv(k)1802 3995 y Fo(=)28 b Fn(u)1962 4010 y Fv(k)2004 3995 y Fn(v)2051 4010 y Fv(k)2126 3995 y Fo(w)m(e)34 b(ha)m(v)m(e)1060 4229 y Fn(u)1116 4248 y Ft(2)1151 4229 y Fi(`)1181 4248 y Fu(\001)p Ft(3)1267 4229 y Fo(=)28 b Fn(u)1427 4178 y Ft(\(2)1489 4154 y Fi(`)p Fh(\000)p Fj(2)1598 4178 y Ft(\))1427 4253 y(12)1629 4229 y Fn(u)1685 4248 y Ft(2)1720 4229 y Fi(`)p Fh(\000)p Fj(2)36 b Fo(=)28 b Fn(u)2020 4178 y Ft(\(2)2082 4154 y Fi(`)p Fh(\000)p Fj(2)2190 4178 y Ft(\))2020 4253 y(12)2222 4229 y Fn(v)2269 4248 y Ft(2)2304 4229 y Fi(`)p Fh(\000)p Fj(3)9 b Fn(v)2464 4248 y Ft(2)2499 4229 y Fi(`)p Fh(\000)p Fj(4)26 b Fn(:)17 b(:)g(:)f(v)2807 4244 y Ft(2)2846 4229 y Fn(v)2893 4244 y Ft(1)2933 4229 y Fn(:)60 4461 y Fo(So)32 b(if)h(2)334 4425 y Fv(`)394 4461 y Fm(j)28 b Fn(u)506 4480 y Ft(2)541 4462 y Fi(`)570 4480 y Fu(\001)p Ft(3)662 4461 y Fo(then)33 b(2)933 4425 y Fv(`)994 4461 y Fm(j)27 b Fn(u)1105 4410 y Ft(\(2)1167 4387 y Fi(`)p Fh(\000)p Fj(2)1276 4410 y Ft(\))1105 4486 y(12)1340 4461 y Fo(so,)33 b(b)m(y)g(Lemma)p 0 .5 0 TeXcolorrgb 1970 4461 a SDict begin H.S end 1970 4461 a 0 .5 0 TeXcolorrgb Fo(9)p 0 .5 0 TeXcolorrgb 2019 4397 a SDict begin H.R end 2019 4397 a 2019 4461 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 2019 4461 a Black Fo(\(i\),)g(2)2231 4425 y Fv(`)p Fu(\000)p Ft(1)2381 4461 y Fm(j)28 b Fn(u)2493 4410 y Ft(\(2)2555 4387 y Fi(`)p Fh(\000)p Fj(2)2663 4410 y Ft(\))2493 4486 y(6)2695 4461 y Fo(.)43 b(Hence)1443 4695 y(2)1492 4653 y Fv(`)p Fu(\000)p Ft(1)1643 4695 y Fm(j)27 b Fn(u)1754 4644 y Ft(\(2)1816 4620 y Fi(`)p Fh(\000)p Fj(2)1925 4644 y Ft(\))1754 4719 y(6)1957 4695 y Fn(u)2013 4714 y Ft(2)2048 4695 y Fi(`)p Fh(\000)p Fj(2)36 b Fo(=)27 b Fn(u)2347 4714 y Ft(2)2382 4695 y Fi(`)p Fh(\000)p Fj(1)2490 4714 y Fu(\001)p Ft(3)2549 4695 y Fn(:)60 4906 y Fo(Also,)37 b(if)e(3)d Fm(j)g Fn(u)600 4925 y Ft(2)635 4906 y Fi(`)665 4925 y Fu(\001)p Ft(3)759 4906 y Fo(where)37 b Fn(`)32 b Fm(\025)h Fo(3)i(then)h(3)c Fm(j)g Fn(u)1733 4925 y Ft(2)1768 4906 y Fi(`)p Fh(\000)p Fj(1)1876 4925 y Fu(\001)p Ft(3)1935 4906 y Fo(,)k(b)m(y)h(Lemma)p 0 .5 0 TeXcolorrgb 2488 4906 a SDict begin H.S end 2488 4906 a 0 .5 0 TeXcolorrgb Fo(9)p 0 .5 0 TeXcolorrgb 2536 4842 a SDict begin H.R end 2536 4842 a 2536 4906 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 2536 4906 a Black Fo(\(ii\).)52 b(Th)m(us)37 b(w)m(e)f(ha)m(v)m (e)h(pro)m(v)m(ed)g(that)e(if)60 5026 y Fn(`)28 b Fm(\025)g Fo(3)k(and)h(2)554 4990 y Fv(`)609 5026 y Fm(\001)21 b Fo(3)28 b Fm(2)g Fn(S)38 b Fo(then)33 b(2)1198 4990 y Fv(`)p Fu(\000)p Ft(1)1344 5026 y Fm(\001)21 b Fo(3)28 b Fm(2)g Fn(S)6 b Fo(.)43 b(Then)34 b(12)27 b Fm(2)h Fn(S)39 b Fo(follo)m(ws.)p 3889 5026 4 66 v 3893 4964 59 4 v 3893 5026 V 3951 5026 4 66 v 60 5112 a SDict begin H.S end 60 5112 a 60 5112 a SDict begin 14.5 H.A end 60 5112 a 60 5112 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.11) cvn H.B /DEST pdfmark end 60 5112 a Black 1986 5289 a Fo(5)p Black eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.6) cvn H.B /DEST pdfmark end 60 -300 a Black Black Black 100 x Fk(Prop)s(osition)38 b(11.)p Black 41 w 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(theorem.1) cvn H.B /ANN pdfmark end 1696 2870 a Black Fp(.)60 3089 y Fo(T)-8 b(o)37 b(pro)m(v)m(e)i(part)e(\(a\),)h(tak)m(e)g Fn(n)e Fm(2)h Fn(S)43 b Fo(and)37 b Fn(p)g Fo(prime.)59 b(First)37 b(note)h(that,)g(from)f(\()p 0 .5 0 TeXcolorrgb 2988 3089 a SDict begin H.S end 2988 3089 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 3037 3025 a SDict begin H.R end 3037 3025 a 3037 3089 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 3037 3089 a Black Fo(\),)h Fn(u)3196 3104 y Fv(np)3314 3089 y Fo(=)e Fn(u)3482 3038 y Ft(\()p Fv(n)p Ft(\))3482 3099 y Fv(p)3583 3089 y Fn(u)3639 3104 y Fv(n)3686 3089 y Fo(.)57 b(No)m(w)60 3210 y(supp)s(ose)48 b(that)e Fn(np)52 b Fm(j)f Fn(u)960 3225 y Fv(np)1042 3210 y Fo(.)86 b(Then)47 b(either)h Fn(p)j Fm(j)g Fn(u)1949 3225 y Fv(n)1996 3210 y Fo(,)f(or,)g(b)m(y)d(Lemma)p 0 .5 0 TeXcolorrgb 2748 3210 a SDict begin H.S end 2748 3210 a 0 .5 0 TeXcolorrgb Fo(5)p 0 .5 0 TeXcolorrgb 2797 3145 a SDict begin H.R end 2797 3145 a 2797 3210 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5) cvn H.B /ANN pdfmark end 2797 3210 a Black Fo(,)j(w)m(e)e(ha)m(v)m(e)g Fn(p)j Fm(j)g Fo(\001)3531 3173 y Ft(\()p Fv(n)p Ft(\))3634 3210 y Fo(,)f(where)60 3330 y(\001)141 3294 y Ft(\()p Fv(n)p Ft(\))271 3330 y Fo(=)27 b(\()p Fn(\013)475 3294 y Fv(n)544 3330 y Fm(\000)c Fn(\014)705 3294 y Fv(n)751 3330 y Fo(\))789 3294 y Ft(2)856 3330 y Fo(=)28 b Fn(u)1016 3294 y Ft(2)1016 3355 y Fv(n)1062 3330 y Fo(\001.)44 b(Hence)34 b Fn(p)28 b Fm(j)f Fn(u)1692 3345 y Fv(n)1738 3330 y Fo(\001.)206 3464 y(Con)m(v)m(ersely)-8 b(,)51 b(supp)s(ose)c Fn(p)h Fm(j)h Fn(u)1344 3479 y Fv(n)1390 3464 y Fo(\001.)81 b(Then)46 b Fn(p)j Fm(j)f Fo(\001)2101 3428 y Ft(\()p Fv(n)p Ft(\))2203 3464 y Fo(,)g(so)d(that,)j(b)m(y)e(Lemma)p 0 .5 0 TeXcolorrgb 3172 3464 a SDict begin H.S end 3172 3464 a 0 .5 0 TeXcolorrgb Fo(5)p 0 .5 0 TeXcolorrgb 3221 3400 a SDict begin H.R end 3221 3400 a 3221 3464 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.5) cvn H.B /ANN pdfmark end 3221 3464 a Black Fo(,)i Fn(p)h Fm(j)f Fn(u)3526 3413 y Ft(\()p Fv(n)p Ft(\))3526 3474 y Fv(p)3628 3464 y Fo(,)g(giving)60 3598 y Fn(pn)28 b Fm(j)f Fn(u)306 3547 y Ft(\()p Fv(n)p Ft(\))306 3608 y Fv(p)408 3598 y Fn(u)464 3613 y Fv(n)538 3598 y Fo(=)g Fn(u)697 3613 y Fv(np)779 3598 y Fo(.)206 3718 y(T)-8 b(o)39 b(pro)m(v)m(e)i(\(b\),)g(tak)m(e)e Fn(n)g Fm(2)h Fn(S)6 b Fo(,)40 b Fn(n)f Fm(6)p Fo(=)g(1)p Fn(;)17 b Fo(6)38 b(or)h(12.)62 b(If)39 b(3)g Fm(2)g Fn(S)45 b Fo(then)39 b(3)p Fn(=)p Fo(3)g(=)f(1)h Fm(2)g Fn(S)6 b Fo(.)63 b(Otherwise,)42 b(b)m(y)60 3839 y(Theorem)p 0 .5 0 TeXcolorrgb 479 3839 a SDict begin H.S end 479 3839 a 0 .5 0 TeXcolorrgb Fo(3)p 0 .5 0 TeXcolorrgb 528 3774 a SDict begin H.R end 528 3774 a 528 3839 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3) cvn H.B /ANN pdfmark end 528 3839 a Black 40 w Fo(and)e(Prop)s(osition)p 0 .5 0 TeXcolorrgb 1298 3839 a SDict begin H.S end 1298 3839 a 0 .5 0 TeXcolorrgb Fo(10)p 0 .5 0 TeXcolorrgb 1395 3774 a SDict begin H.R end 1395 3774 a 1395 3839 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.10) cvn H.B /ANN pdfmark end 1395 3839 a Black Fo(,)i(w)m(e)f(ha)m(v)m(e)h Fn(n=p)e Fm(2)h Fn(S)46 b Fo(for)39 b(some)i(prime)g(factor)f Fn(p)g Fo(of)f Fn(n)p Fo(.)66 b(Th)m(us)42 b(w)m(e)60 3959 y(obtain)33 b(a)f(sequence)j Fn(n;)17 b(n=p;)g Fo(\()p Fn(n=p)p Fo(\))p Fn(=p)1481 3923 y Fu(0)1504 3959 y Fn(;)g(:)g(:)g(:)48 b Fo(of)32 b(elemen)m(ts)j(of)e Fn(S)6 b Fo(,)32 b(whic)m(h)i(stops)g (only)f(at)f(1,)h(6)f(or)h(12.)43 b(But)60 4079 y(clearly)32 b(6)f(and)g(12)g(cannot)g(b)s(oth)g(b)s(e)g(basic,)i(so)e(the)h(pro)s (cess)g(will)g(stop)f(at)g(either)h(1)f(\(alw)m(a)m(ys)h(basic!\))44 b(or)60 4200 y(at)33 b(most)h(one)f(of)g(6)g(and)g(12.)45 b(This)34 b(sho)m(ws)h(that)e(this)h(sequence,)i(written)e(bac)m(kw)m 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b(\(c\),)i(w)m(e)h(kno)m(w)f(from)f(Theorem)p 0 .5 0 TeXcolorrgb 2172 643 a SDict begin H.S end 2172 643 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 2269 579 a SDict begin H.R end 2269 579 a 2269 643 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 2269 643 a Black Fo(\(a\))g(that)g(for)g(2)d Fm(2)g(P)3038 658 y Fv(T)6 b(;)p Ft(2)p Fv(n)3221 643 y Fo(all)34 b(prime)h(divisors) 60 763 y(of)e(2)p Fn(n)h Fo(are)g(sp)s(ecial,)i(so)e(2)f(is)i(sp)s (ecial.)48 b(Con)m(v)m(ersely)-8 b(,)38 b(if)33 b(2)h(is)g(sp)s(ecial,) i(then)e(all)g(prime)h(factors)f(of)f(2)p Fn(n)h Fo(are)60 884 y(sp)s(ecial,)g(and)e Fn(P)46 b Fo(is)33 b(ev)m(en,)i(so)d(that,)h (b)m(y)g(Theorem)p 0 .5 0 TeXcolorrgb 1956 884 a SDict begin H.S end 1956 884 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 2054 819 a SDict begin H.R end 2054 819 a 2054 884 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 2054 884 a Black Fo(\(a\),)f(2)c Fm(2)g(P)2478 899 y Fv(T)6 b(;)p Ft(2)p Fv(n)2627 884 y Fo(.)p 3889 884 4 66 v 3893 821 59 4 v 3893 884 V 3951 884 4 66 v 60 985 a SDict begin H.S end 60 985 a 60 985 a SDict begin 14.5 H.A end 60 985 a 60 985 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.14) cvn H.B /DEST pdfmark end 60 985 a Black 97 x Fk(Corollary)39 b(14.)p Black 42 w Fl(If)34 b Fn(n)28 b Fm(2)g Fn(T)49 b Fl(and)p Black 205 1281 a Fm(\017)p Black 49 w Fl(al)5 b(l)34 b(o)-5 b(dd)35 b(prime)f(factors)g(of)h Fn(m)g Fl(divide)f Fn(v)1791 1296 y Fv(n)1838 1281 y Fl(;)304 1442 y(and)p Black 205 1644 a Fm(\017)p Black 49 w Fl(if)h Fn(m)g Fl(is)f(even)g(then)h(every) g(prime)f(divisor)g(of)h Fn(nm)g Fl(is)g(sp)-5 b(e)g(cial;)60 1843 y(then)35 b Fn(nm)28 b Fm(2)g Fn(T)14 b Fl(.)p Black 60 2041 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(On)29 b(successiv)m(ely)j(m)m (ultiplying)e Fn(n)f Fo(b)m(y)g(\014rst)g(the)g(o)s(dd)f(and)g(then)h (the)g(ev)m(en)h(prime)f(divisors)h(of)e Fn(m)p Fo(,)60 2162 y(w)m(e)38 b(see)g(from)f(Theorem)p 0 .5 0 TeXcolorrgb 1022 2162 a SDict begin H.S end 1022 2162 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 1119 2097 a SDict begin H.R end 1119 2097 a 1119 2162 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 1119 2162 a Black Fo(\(a\))g(that)g(the)g(stated)h(conditions)g(ensure)h (that)e(w)m(e)h(sta)m(y)g(within)g Fn(T)51 b Fo(while)60 2282 y(doing)33 b(this.)p 3889 2282 4 66 v 3893 2219 59 4 v 3893 2282 V 3951 2282 4 66 v 206 2475 a(This)43 b(result)f(extends)i(Theorem)e(5\(i\))f(of)g(Somer)h([)p 0 .5 0 TeXcolorrgb 2150 2475 a SDict begin H.S end 2150 2475 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 2248 2411 a SDict begin H.R end 2248 2411 a 2248 2475 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1393479) cvn H.B /ANN pdfmark end 2248 2475 a Black Fo(],)i(whic)m(h)f(has)e(the)h(condition)g (that)g(`)p Fn(m)f Fo(is)h(a)60 2596 y(pro)s(duct)33 b(of)f(sp)s(ecial)h(primes)h(or)f(divides)h Fn(n)p Fo(')f(instead)g(of) f(`all)h(o)s(dd)f(prime)i(factors)e(of)h Fn(m)f Fo(divide)i Fn(v)3697 2611 y Fv(n)3744 2596 y Fo('.)60 2765 y SDict begin H.S end 60 2765 a 60 2765 a SDict begin 14.5 H.A end 60 2765 a 60 2765 a SDict begin [ /View [/XYZ H.V] /Dest (section.5) cvn H.B /DEST pdfmark end 60 2765 a 163 x Fp(5)161 b(Preliminary)54 b(results)e(for)i Fg(T)20 b Fp(.)60 3147 y Fo(W)-8 b(e)33 b(\014rst)h(quote)f(the)h(imp)s(ortan)m (t)f(result)h(of)e(Somer)i(for)e Fn(T)14 b Fo(,)33 b(corresp)s(onding)h (to)e(his)i(result)g(\(Theorem)p 0 .5 0 TeXcolorrgb 3911 3147 a SDict begin H.S end 3911 3147 a 0 .5 0 TeXcolorrgb Fo(3)p 0 .5 0 TeXcolorrgb 3960 3083 a SDict begin H.R end 3960 3083 a 3960 3147 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3) cvn H.B /ANN pdfmark end 3960 3147 a Black 60 3267 a Fo(ab)s(o)m(v)m(e\))f(for)f Fn(S)6 b Fo(.)60 3292 y SDict begin H.S end 60 3292 a 60 3292 a SDict begin 14.5 H.A end 60 3292 a 60 3292 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.15) cvn H.B /DEST pdfmark end 60 3292 a Black 174 x Fk(Theorem)39 b(15)g Fo(\(Somer)34 b([)p 0 .5 0 TeXcolorrgb 1051 3466 a SDict begin H.S end 1051 3466 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 1149 3402 a SDict begin H.R end 1149 3402 a 1149 3466 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1393479) cvn H.B /ANN pdfmark end 1149 3466 a Black Fo(,)g(Theorem)h(5]\))p Fk(.)p Black 42 w Fl(The)-5 b(or)g(em)p 0 .5 0 TeXcolorrgb 2215 3466 a SDict begin H.S end 2215 3466 a 0 .5 0 TeXcolorrgb Fl(3)p 0 .5 0 TeXcolorrgb 2265 3401 a SDict begin H.R end 2265 3401 a 2265 3466 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.3) cvn H.B /ANN pdfmark end 2265 3466 a Black 36 w Fl(holds)35 b(with)h(the)g(set)g Fn(S)41 b Fl(r)-5 b(eplac)g(e)g(d)35 b(by)h(the)g(set)60 3586 y Fn(T)14 b Fl(.)206 3785 y Fo(Jarden)36 b([)p 0 .5 0 TeXcolorrgb 557 3785 a SDict begin H.S end 557 3785 a 0 .5 0 TeXcolorrgb Fo(13)p 0 .5 0 TeXcolorrgb 655 3721 a SDict begin H.R end 655 3721 a 655 3785 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0197383) cvn H.B /ANN pdfmark end 655 3785 a Black Fo(,)h(Theorem)f(E])g(pro)m(v)m(ed)i(this)e(result)g(for)f(the) h(classical)i(Lucas)e(sequence)i(\(i.e.,)f Fn(P)46 b Fo(=)33 b(1,)60 3905 y Fn(Q)28 b Fo(=)f Fm(\000)p Fo(1\))33 b(under)h(the)f(restriction)g Fn(p)1425 3920 y Ft(max)1588 3905 y Fm(6)p Fo(=)28 b(3.)60 3930 y SDict begin H.S end 60 3930 a 60 3930 a SDict begin 14.5 H.A end 60 3930 a 60 3930 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.16) cvn H.B /DEST pdfmark end 60 3930 a Black 174 x Fk(Lemma)39 b(16.)p Black 42 w Fl(Supp)-5 b(ose)34 b Fn(q)39 b Fl(is)34 b(a)h(sp)-5 b(e)g(cial)34 b(prime.)44 b(Then)34 b Fn(q)2214 4068 y Fv(e)2247 4076 y Fi(n)2321 4104 y Fm(j)27 b Fn(v)2423 4119 y Fv(n)2470 4104 y Fl(,)35 b(wher)-5 b(e)34 b Fn(e)2855 4119 y Fv(n)2930 4104 y Fm(\025)28 b(b)p Fo(log)3206 4128 y Fv(q)3260 4104 y Fn(n)p Fm(c)p Fl(.)p Black 60 4303 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(F)d(rom)32 b(the)h(recurrence,)i (it)e(is)g(easy)g(to)g(see)g(that)g(w)m(e)g(can)g(tak)m(e)1432 4599 y Fn(e)1477 4614 y Fv(n)1552 4599 y Fo(=)1655 4429 y Ff(\()1735 4450 y(\004)1792 4491 y Fv(n)p 1792 4508 43 4 v 1796 4565 a Ft(2)1845 4450 y Ff(\005)1914 4531 y Fo(+)22 b(1)p Fn(;)49 b Fo(if)33 b Fn(q)e Fo(=)d(2;)1735 4594 y Ff(\004)1792 4636 y Fv(n)p Ft(+1)p 1792 4652 133 4 v 1841 4709 a(2)1935 4594 y Ff(\005)1999 4675 y Fn(;)49 b Fo(if)33 b Fn(q)e Fm(\025)d Fo(3)p Fn(;)60 4911 y Fo(the)37 b(sligh)m(tly)h(higher)f(v)-5 b(alue)37 b(for)e Fn(q)k Fo(=)34 b(2)i(coming)h(from)f(the)h(fact)f(that)g Fn(v)2769 4926 y Ft(0)2843 4911 y Fo(=)e(2.)55 b(Then)38 b(use)3515 4830 y Ff(\004)3562 4911 y Fo(log)3688 4934 y Fv(q)3743 4911 y Fn(n)3801 4830 y Ff(\005)3883 4911 y Fm(\024)60 4959 y Ff(\004)117 5001 y Fv(n)p Ft(+1)p 117 5017 V 166 5074 a(2)260 4959 y Ff(\005)307 5040 y Fo(.)p 3889 5040 4 66 v 3893 4977 59 4 v 3893 5040 V 3951 5040 4 66 v Black 1986 5289 a(8)p Black eop end end %%Page: 9 9 TeXDict begin HPSdict begin 9 8 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.9) cvn H.B /DEST pdfmark end 60 -300 a Black Black 206 -200 a Fo(W)-8 b(e)33 b(then)g(immediately)i(obtain)d(the)h(follo)m (wing.)60 -181 y SDict begin H.S end 60 -181 a 60 -181 a SDict begin 14.5 H.A end 60 -181 a 60 -181 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.17) cvn H.B /DEST pdfmark end 60 -181 a Black 184 x Fk(Corollary)i(17)e Fo(\(Sp)s(ecial)c(case)h (of)e(Somer)h([)p 0 .5 0 TeXcolorrgb 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327 a Black 203 x Fk(Prop)s(osition)38 b(18.)p Black 41 w Fl(If)d Fn(`)27 b Fm(\025)h Fo(2)35 b Fl(and)f Fo(2)1448 494 y Fv(`)1503 530 y Fm(\001)22 b Fo(3)27 b Fm(2)i Fn(T)14 b Fl(,)34 b(then)h Fo(2)2125 494 y Fv(`)2185 530 y Fm(2)28 b Fn(T)14 b Fl(.)p Black 60 734 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(Put)39 b Fn(L)e Fo(=)g(2)827 697 y Fv(`)859 734 y Fo(.)60 b(If)38 b(2)f(is)i(sp)s(ecial,)h(then,)g(b)m(y)f(Corollary)p 0 .5 0 TeXcolorrgb 2423 734 a SDict begin H.S end 2423 734 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 2520 669 a SDict begin H.R end 2520 669 a 2520 734 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.17) cvn H.B /ANN pdfmark end 2520 734 a Black Fo(,)h Fn(L)d Fm(2)g Fn(T)52 b Fo(for)37 b(all)h Fn(`)e Fm(\025)i Fo(1.)59 b(So)38 b(w)m(e)h(can)60 854 y(assume)34 b(that)e(2)g(is)h(not)f(sp)s(ecial.)45 b(W)-8 b(e)32 b(then)h(kno)m(w)h(that)e Fn(Q)h Fo(m)m(ust)g(b)s(e)g(o)s (dd,)f(as)h(if)f(it)g(w)m(ere)i(ev)m(en)g(then)f(w)m(e)60 974 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Fl(for)f(which)g(the)h(r)-5 b(ank)35 b(of)f(app)-5 b(e)g(ar)g(anc)g(e) 34 b Fn(!)t Fo(\()p Fn(p)p Fo(\))g Fl(of)g Fn(p)h Fl(\(in)f Fo(\()p Fn(u)2966 2314 y Fv(n)3013 2299 y Fo(\))p Fl(\))g(is)h(even;)p Black 205 2502 a Fm(\017)p Black 49 w Fl(the)g(sp)-5 b(e)g(cial)34 b(primes;)p Black 205 2705 a Fm(\017)p Black 49 w Fl(the)h(prime)f Fo(2)p Fl(,)h(unless)f Fn(P)48 b Fl(is)35 b(o)-5 b(dd)34 b(and)g Fn(Q)i Fl(is)e(even.)p Black 60 2909 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(T)d(ak)m(e)48 b(a)e(prime)h Fn(p)f Fo(with)h Fn(p)k Fe(-)g Fo(2)p Fn(Q)p Fo(,)f(and)c(let)h Fn(!)54 b Fo(=)d Fn(!)t Fo(\()p Fn(p)p Fo(\).)84 b(If)47 b Fn(!)i Fo(is)e(ev)m(en,)k(then)c(the)g(iden)m(tit)m (y)60 3029 y Fn(u)116 3044 y Ft(2)p Fv(n)225 3029 y Fo(=)28 b Fn(u)385 3044 y Fv(n)431 3029 y Fn(v)478 3044 y Fv(n)550 3029 y Fo(for)23 b Fn(n)28 b Fo(=)f Fn(!)t(=)p Fo(2)c(sho)m(ws)j(that)e Fn(p)j Fm(j)h Fn(v)1719 3044 y Fv(n)1766 3029 y Fo(,)d Fn(p)j Fm(2)g(P)2058 3044 y Ft(2nd)2176 3029 y Fo(.)41 b(The)25 b(iden)m(tit)m(y)h(also)e(sho)m(ws)h(that)f Fm(P)3518 3044 y Ft(2nd)3664 3029 y Fm(\032)k(P)3838 3044 y Ft(1st)3933 3029 y Fo(.)206 3150 y(Con)m(v)m(ersely)-8 b(,)33 b(if)28 b Fn(p)g Fm(2)g(P)1041 3165 y Ft(2nd)1159 3150 y Fo(,)i(sa)m(y)f Fn(p)f Fm(j)f Fn(v)1559 3165 y Fv(n)1606 3150 y Fo(,)j(then)f Fn(p)f Fm(j)f Fn(u)2069 3165 y Ft(2)p Fv(n)2151 3150 y Fo(,)i(so)g(that,)g(b)m(y)h([)p 0 .5 0 TeXcolorrgb 2717 3150 a SDict begin H.S end 2717 3150 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 2814 3085 a SDict begin H.R end 2814 3085 a 2814 3150 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 2814 3150 a Black Fo(,)g(Prop)s(osition)f(1\(iv\)],)h Fn(!)h Fm(j)c Fo(2)p Fn(n)p Fo(.)60 3270 y(Ho)m(w)m(ev)m(er,)35 b(from)d(the)h(iden)m(tit)m(y)1239 3150 y SDict begin H.S end 1239 3150 a 1239 3150 a SDict begin 14.5 H.A end 1239 3150 a 1239 3150 a SDict begin [ /View [/XYZ H.V] /Dest (equation.6) cvn H.B /DEST pdfmark end 1239 3150 a 1645 3390 a Fn(u)1701 3349 y Ft(2)1701 3415 y Fv(n)1769 3390 y Fm(\000)23 b Fo(\001)p Fn(v)2001 3349 y Ft(2)1997 3415 y Fv(n)2072 3390 y Fo(=)k(4)p Fn(Q)2301 3349 y Fv(n)2348 3390 y Fn(;)1460 b Fo(\(6\))60 3565 y(w)m(e)34 b(ha)m(v)m(e)g Fn(p)27 b Fe(-)h Fn(u)617 3580 y Fv(n)663 3565 y Fo(,)33 b(so)f(that)h Fn(!)j Fo(is)d(ev)m(en.)206 3685 y(No)m(w)h(tak)m(e)g(a)f (prime)h Fn(p)g Fo(with)f Fn(p)c Fm(j)g Fn(Q)p Fo(.)46 b(Then)34 b(from)f Fn(v)2123 3700 y Fv(n)2199 3685 y Fm(\021)c Fn(P)2382 3649 y Fv(n)2473 3685 y Fo(\(mo)s(d)k Fn(p)p Fo(\))g(w)m(e)i(see)f(that,)f(for)g Fn(p)g Fo(to)g(b)s(e)h(in)60 3805 y Fm(P)129 3820 y Ft(2nd)247 3805 y Fo(,)e Fn(p)h Fo(m)m(ust)g(b)s(e)f(sp)s(ecial.)45 b(In)32 b(particular,)h(2)27 b Fm(62)i(P)1955 3820 y Ft(2nd)2105 3805 y Fo(when)k Fn(P)46 b Fo(is)33 b(o)s(dd)f(and)g Fn(Q)g Fo(is)h(ev)m(en.)45 b(F)-8 b(urther,)33 b(if)f Fn(P)60 3926 y Fo(is)h(ev)m(en)h(then)f Fn(v)649 3941 y Ft(1)717 3926 y Fo(=)27 b Fn(P)46 b Fo(is)33 b(ev)m(en,)i(while)e(if)g Fn(P)46 b Fo(and)32 b Fn(Q)h Fo(are)g(b)s(oth)f(o)s(dd)g(then)i Fn(v)2884 3941 y Ft(3)2951 3926 y Fo(=)27 b Fn(P)14 b Fo(\()p Fn(P)3246 3890 y Ft(2)3307 3926 y Fm(\000)22 b Fo(3)p Fn(Q)p Fo(\))33 b(is)g(ev)m(en.)206 4046 y(Finally)-8 b(,)48 b(for)d Fn(p)j Fe(-)g Fo(2)p Fn(Q)p Fo(,)g(c)m(ho)s(ose)e Fn(m)f Fo(o)s(dd,)i(and)e(su\016cien)m (tly)j(large)c(that)h(w)m(e)h(can)f(tak)m(e)g Fn(p)g Fo(to)f(b)s(e)h(a)60 4167 y(primitiv)m(e)34 b(prime)g(divisor)f(of)f Fn(u)1238 4182 y Fv(m)1304 4167 y Fo(.)44 b(Then)34 b(w)m(e)f(ha)m(v)m (e)h Fn(!)t Fo(\()p Fn(p)p Fo(\))27 b(=)h Fn(m)p Fo(,)k(and)h(hence)h Fn(p)28 b Fm(2)g(P)3164 4182 y Ft(1st)3281 4167 y Fm(n)22 b(P)3422 4182 y Ft(2nd)3540 4167 y Fo(.)p 3889 4287 4 66 v 3893 4224 59 4 v 3893 4287 V 3951 4287 4 66 v 206 4482 a(Our)36 b(next)h(lemma)f(is)h(an)e(easy)i(exercise.)56 b(Dic)m(kson)36 b([)p 0 .5 0 TeXcolorrgb 2214 4482 a SDict begin H.S end 2214 4482 a 0 .5 0 TeXcolorrgb Fo(8)p 0 .5 0 TeXcolorrgb 2263 4418 a SDict begin H.R end 2263 4418 a 2263 4482 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0245499) cvn H.B /ANN pdfmark end 2263 4482 a Black Fo(,)h(pp.)f(67)f(and)h(271])f(traces)i(the)f(result)h(bac)m(k)60 4602 y(to)32 b(an)h(`anon)m(ymous)h(writer')f(in)g(1830)f([)p 0 .5 0 TeXcolorrgb 1535 4602 a SDict begin H.S end 1535 4602 a 0 .5 0 TeXcolorrgb Fo(25)p 0 .5 0 TeXcolorrgb 1632 4538 a SDict begin H.R end 1632 4538 a 1632 4602 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.Anon) cvn H.B /ANN pdfmark end 1632 4602 a Black Fo(],)h(and)g(also)f(to)g(Lucas)i([)p 0 .5 0 TeXcolorrgb 2529 4602 a SDict begin H.S end 2529 4602 a 0 .5 0 TeXcolorrgb Fo(15)p 0 .5 0 TeXcolorrgb 2626 4538 a SDict begin H.R end 2626 4538 a 2626 4602 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1505161) cvn H.B /ANN pdfmark end 2626 4602 a Black Fo(,)f(p.)g(229].)60 4627 y SDict begin H.S end 60 4627 a 60 4627 a SDict begin 14.5 H.A end 60 4627 a 60 4627 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.20) cvn H.B /DEST pdfmark end 60 4627 a Black 179 x Fk(Lemma)42 b(20.)p Black 43 w Fl(F)-7 b(or)37 b Fn(p)g Fl(an)g(o)-5 b(dd)37 b(prime)g(and)g Fn(j)h Fo(=)32 b(1)p Fn(;)17 b Fo(2)p Fn(;)g(:)g(:)g(:)e(;)i Fo(\()p Fn(p)24 b Fm(\000)g Fo(1\))p Fn(=)p Fo(2)p Fl(,)38 b(the)f(expr)-5 b(ession)36 b Fn(B)3437 4821 y Fv(j)3506 4806 y Fo(:=)3641 4725 y Ff(\000)3687 4762 y Fv(p)p Fu(\000)p Ft(1)3734 4840 y Fv(j)3813 4725 y Ff(\001)3883 4806 y Fm(\000)60 4941 y Fo(\()p Fm(\000)p Fo(1\))262 4905 y Fv(j)334 4941 y Fl(is)e(divisible)g(by)h Fn(p)p Fl(.)p Black 1986 5289 a Fo(9)p Black eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.10) cvn H.B /DEST pdfmark end 60 -300 a Black Black 206 -200 a Fo(The)50 b(follo)m(wing)f(result)h(essen)m(tially)h(dates)e (bac)m(k)h(to)e(Lucas)i([)p 0 .5 0 TeXcolorrgb 2594 -200 a SDict begin H.S end 2594 -200 a 0 .5 0 TeXcolorrgb Fo(15)p 0 .5 0 TeXcolorrgb 2691 -265 a SDict begin H.R end 2691 -265 a 2691 -200 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1505161) cvn H.B /ANN pdfmark end 2691 -200 a Black Fo(,)j(p.)c(210])f(and)h(Carmic)m(hael)h([)p 0 .5 0 TeXcolorrgb 3884 -200 a SDict begin H.S end 3884 -200 a 0 .5 0 TeXcolorrgb Fo(6)p 0 .5 0 TeXcolorrgb 3933 -265 a SDict begin H.R end 3933 -265 a 3933 -200 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1516755) cvn H.B /ANN pdfmark end 3933 -200 a Black Fo(,)60 -80 y(Theorem)34 b(X].)60 -55 y SDict begin H.S end 60 -55 a 60 -55 a SDict begin 14.5 H.A end 60 -55 a 60 -55 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.21) cvn H.B /DEST pdfmark end 60 -55 a Black 164 x Fk(Lemma)39 b(21.)p Black Black 127 w Fl(\(i\))p Black 49 w(F)-7 b(or)34 b Fn(n)28 b Fm(2)g Fe(N)34 b Fl(and)h(any)g(prime)f Fn(p)p Fl(,)g Fn(p)h Fl(divides)f Fn(v)2579 124 y Fv(np)2697 109 y Fl(if)g(and)h(only)f(if)h Fn(p)g Fl(divides)f Fn(v)3742 124 y Fv(n)3789 109 y Fl(.)p Black 116 307 a(\(ii\))p Black 48 w(F)-7 b(or)34 b Fn(n)28 b Fm(2)g Fe(N)35 b Fl(and)f(any)h(o)-5 b(dd)34 b(prime)g Fn(p)p Fl(,)h Fn(v)1769 322 y Fv(n)1851 307 y Fl(divides)f Fn(v)2222 322 y Fv(np)2339 307 y Fl(and)h Fn(v)2576 322 y Fv(np)2658 307 y Fn(=v)2754 322 y Fv(n)2829 307 y Fm(\021)28 b Fn(v)2985 271 y Fv(p)p Fu(\000)p Ft(1)2981 332 y Fv(n)3159 307 y Fo(\(mo)s(d)33 b Fn(p)p Fo(\))p Fl(.)p Black 60 496 a(Pr)-5 b(o)g(of.)p Black Black 141 w Fo(\(i\))p Black 48 w(No)m(w)33 b Fn(v)878 511 y Ft(2)946 496 y Fo(=)27 b Fn(v)1100 460 y Ft(2)1096 521 y(1)1162 496 y Fm(\000)22 b Fo(2)p Fn(Q)p Fo(,)33 b(whic)m(h)h(is)f(ev)m(en)h(i\013)f Fn(v)2210 511 y Ft(1)2282 496 y Fo(is)g(ev)m(en.)45 b(Also,)33 b(for)f Fn(p)c Fm(\025)g Fo(3,)950 804 y Fn(v)1001 757 y Fv(p)997 828 y Ft(1)1068 804 y Fo(=)g(\()p Fn(\013)23 b Fo(+)f Fn(\014)6 b Fo(\))1492 763 y Fv(p)1558 804 y Fo(=)28 b Fn(v)1709 819 y Fv(p)1771 804 y Fo(+)1869 672 y Ft(\()p Fv(p)p Fu(\000)p Ft(1\))p Fv(=)p Ft(2)1923 709 y Ff(X)1933 919 y Fv(j)t Ft(=1)2104 663 y Ff(\022)2177 736 y Fn(p)2178 872 y(j)2226 663 y Ff(\023)2299 804 y Fn(Q)2376 763 y Fv(j)2413 804 y Fn(v)2460 819 y Fv(p)p Fu(\000)p Ft(2)p Fv(j)2650 804 y Fm(\021)g Fn(v)2802 819 y Fv(p)2942 804 y Fo(\(mo)s(d)33 b Fn(p)p Fo(\))p Fn(:)3835 659 y SDict begin H.S end 3835 659 a 3835 659 a SDict begin 17.5 H.A end 3835 659 a 3835 659 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 3835 659 a 145 x Fo(\(7\))304 1102 y(No)m(w)g(replace)h Fn(\013)q(;)17 b(\014)37 b Fo(b)m(y)d Fn(\013)1255 1066 y Fv(n)1301 1102 y Fn(;)17 b(\014)1406 1066 y Fv(n)1453 1102 y Fo(.)p Black 125 1301 a(\(ii\))p Black 49 w(T)-8 b(aking)33 b Fn(p)g Fo(o)s(dd)f(and)h Fn(B)1167 1316 y Fv(j)1236 1301 y Fo(de\014ned)h(as)f(in)f(Lemma)p 0 .5 0 TeXcolorrgb 2153 1301 a SDict begin H.S end 2153 1301 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 2251 1236 a SDict begin H.R end 2251 1236 a 2251 1301 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.20) cvn H.B /ANN pdfmark end 2251 1301 a Black Fo(,)h(w)m(e)g(ha)m(v)m(e)940 1503 y Fn(v)987 1518 y Fv(p)1055 1503 y Fo(=)27 b(\()p Fn(\013)c Fo(+)f Fn(\014)6 b Fo(\)\()p Fn(\013)1579 1462 y Fv(p)p Fu(\000)p Ft(1)1730 1503 y Fm(\000)23 b Fn(\013)1893 1462 y Fv(p)p Fu(\000)p Ft(2)2022 1503 y Fn(\014)28 b Fo(+)22 b Fm(\001)17 b(\001)g(\001)j Fo(+)i Fn(\014)2500 1462 y Fv(p)p Fu(\000)p Ft(1)2630 1503 y Fo(\))1055 1754 y(=)27 b(\()p Fn(\013)c Fo(+)f Fn(\014)6 b Fo(\)\(\()p Fn(\013)22 b Fo(+)g Fn(\014)6 b Fo(\))1835 1713 y Fv(p)p Fu(\000)p Ft(1)1987 1754 y Fm(\000)2095 1626 y Fv(p)p Fu(\000)p Ft(2)2086 1659 y Ff(X)2097 1869 y Fv(j)t Ft(=1)2247 1754 y Fn(B)2321 1769 y Fv(j)2357 1754 y Fn(\013)2420 1713 y Fv(p)p Fu(\000)p Ft(1)p Fu(\000)p Fv(j)2637 1754 y Fn(\014)2698 1713 y Fv(j)2734 1754 y Fo(\))1055 2130 y(=)27 b Fn(v)1205 2145 y Ft(1)1261 1930 y Ff(0)1261 2109 y(@)1348 2130 y Fn(v)1399 2083 y Fv(p)p Fu(\000)p Ft(1)1395 2154 y(1)1551 2130 y Fm(\000)1651 1997 y Ft(\()p Fv(p)p Fu(\000)p Ft(3\))p Fv(=)p Ft(2)1704 2035 y Ff(X)1715 2245 y Fv(j)t Ft(=1)1885 2130 y Fn(B)1959 2145 y Fv(j)1996 2130 y Fn(Q)2073 2089 y Fv(j)2110 2130 y Fn(v)2157 2145 y Fv(p)p Fu(\000)p Ft(1)p Fu(\000)p Ft(2)p Fv(j)2431 2130 y Fm(\000)c Fn(B)2605 2145 y Ft(\()p Fv(p)p Fu(\000)p Ft(1\))p Fv(=)p Ft(2)2860 2130 y Fn(Q)2937 2089 y Ft(\()p Fv(p)p Fu(\000)p Ft(1\))p Fv(=)p Ft(2)3193 1930 y Ff(1)3193 2109 y(A)3297 2130 y Fn(:)304 2444 y Fo(so)33 b(that)f(the)h(result)h (of)e Fn(p)g Fo(o)s(dd)g(follo)m(ws)i(b)m(y)f(replacing)h Fn(\013)q(;)17 b(\014)37 b Fo(b)m(y)d Fn(\013)2731 2408 y Fv(n)2777 2444 y Fn(;)17 b(\014)2882 2408 y Fv(n)2961 2444 y Fo(and)33 b(using)g(Lemma)p 0 .5 0 TeXcolorrgb 3754 2444 a SDict begin H.S end 3754 2444 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 3852 2380 a SDict begin H.R end 3852 2380 a 3852 2444 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.20) cvn H.B /ANN pdfmark end 3852 2444 a Black Fo(.)p 3889 2565 4 66 v 3893 2502 59 4 v 3893 2565 V 3951 2565 4 66 v 60 2712 a SDict begin H.S end 60 2712 a 60 2712 a SDict begin 14.5 H.A end 60 2712 a 60 2712 a SDict begin [ /View [/XYZ H.V] /Dest (section.6) cvn H.B /DEST pdfmark end 60 2712 a 183 x Fp(6)161 b(Pro)t(of)54 b(of)g(Theorem)p 0 .5 0 TeXcolorrgb 1615 2895 a SDict begin H.S end 1615 2895 a 0 .5 0 TeXcolorrgb Fp(12)p 0 .5 0 TeXcolorrgb 1777 2802 a SDict begin H.R end 1777 2802 a 1777 2895 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 1777 2895 a Black 60 3114 a Fo(W)-8 b(e)32 b(no)m(w)g(pro)m(v)m(e)g(part)f(\(a\))g(of)g (Theorem)p 0 .5 0 TeXcolorrgb 1576 3114 a SDict begin H.S end 1576 3114 a 0 .5 0 TeXcolorrgb Fo(12)p 0 .5 0 TeXcolorrgb 1674 3050 a SDict begin H.R end 1674 3050 a 1674 3114 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.12) cvn H.B /ANN pdfmark end 1674 3114 a Black Fo(.)43 b(First)31 b(tak)m(e)h Fn(p)f Fo(o)s(dd)g(and)h Fn(n)c Fm(2)g Fn(T)14 b Fo(.)43 b(Then,)32 b(b)m(y)h(Lemma)p 0 .5 0 TeXcolorrgb 3732 3114 a SDict begin H.S end 3732 3114 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 3830 3050 a SDict begin H.R end 3830 3050 a 3830 3114 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.21) cvn H.B /ANN pdfmark end 3830 3114 a Black Fo(\(i\),)60 3234 y(if)h Fn(p)c Fe(-)g Fn(v)335 3249 y Fv(n)416 3234 y Fo(then)35 b Fn(p)30 b Fe(-)g Fn(v)824 3249 y Fv(np)907 3234 y Fo(,)k(so)h Fn(np)30 b Fm(62)h Fn(T)14 b Fo(.)48 b(Con)m(v)m(ersely)-8 b(,)37 b(if)d Fn(p)2124 3198 y Fv(\025)2170 3234 y Fm(k)p Fn(v)2267 3249 y Fv(n)2348 3234 y Fo(for)f(some)i Fn(\025)30 b Fm(\025)h Fo(1)j(then)h(b)m(y)g(Lemma)p 0 .5 0 TeXcolorrgb 3732 3234 a SDict begin H.S end 3732 3234 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 3830 3170 a SDict begin H.R end 3830 3170 a 3830 3234 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.21) cvn H.B /ANN pdfmark end 3830 3234 a Black Fo(\(ii\))60 3355 y Fn(p)109 3318 y Fv(\025)p Ft(+1)272 3355 y Fm(j)27 b Fn(v)374 3370 y Fv(np)457 3355 y Fo(.)44 b(Since)33 b Fn(n)28 b Fm(j)g Fn(v)971 3370 y Fv(n)1050 3355 y Fo(and)33 b Fn(pv)1336 3370 y Fv(n)1411 3355 y Fm(j)27 b Fn(v)1513 3370 y Fv(np)1628 3355 y Fo(w)m(e)34 b(ha)m(v)m(e)g Fn(np)28 b Fm(2)g Fn(T)14 b Fo(.)206 3475 y(No)m(w)36 b(tak)m(e)g Fn(p)c Fo(=)g(2,)k(and)f(supp)s (ose)h(that)f(b)s(oth)g Fn(n)h Fo(and)f(2)p Fn(n)g Fo(are)g(in)h Fn(T)14 b Fo(.)51 b(First)35 b(note)g(that)g Fn(v)3533 3490 y Fv(n)3615 3475 y Fo(m)m(ust)i(b)s(e)60 3595 y(ev)m(en,)d(as)e (otherwise)i Fn(v)909 3610 y Ft(2)p Fv(n)1019 3595 y Fo(=)28 b Fn(v)1174 3559 y Ft(2)1170 3620 y Fv(n)1238 3595 y Fm(\000)22 b Fo(2)p Fn(Q)1463 3559 y Fv(n)1542 3595 y Fo(w)m(ould)34 b(b)s(e)e(o)s(dd.)43 b(Also,)33 b(w)m(e)g(ha)m(v)m(e)h Fn(n)28 b Fm(j)f Fn(Q)3022 3559 y Fv(n)3069 3595 y Fo(,)33 b(so)f(that)g(ev)m(ery)i(prime)60 3716 y(factor)k Fn(q)k Fo(of)c Fn(n)h Fo(divides)h Fn(Q)p Fo(.)62 b(\(Note)38 b(that)g(this)h(w)m(orks)h(to)s(o)e(if)g Fn(q)k Fo(=)37 b(2,)j(as)f(then)g(4)e Fm(j)h Fn(v)3258 3731 y Ft(2)p Fv(n)3340 3716 y Fo(.\))61 b(But)39 b Fn(q)j Fo(m)m(ust)60 3836 y(also)33 b(divide)i Fn(P)14 b Fo(,)33 b(as)h(otherwise)h Fn(v)1287 3851 y Fv(n)1363 3836 y Fm(\021)29 b Fn(P)1546 3800 y Fv(n)1621 3836 y Fm(6\021)h Fo(0)44 b(\(mo)s(d)33 b Fn(q)t Fo(\).)46 b(Hence)34 b Fn(q)j Fo(is)d(sp)s(ecial,)h(and)e Fn(n)h Fo(is)g(a)f(pro)s(duct)g(of) 60 3957 y(sp)s(ecial)j(primes.)54 b(If)35 b Fn(n)g Fo(is)h(ev)m(en,)i (then)e(2)f(is)h(sp)s(ecial,)h(so)f Fn(P)48 b Fo(and)36 b Fn(Q)f Fo(are)h(b)s(oth)f(ev)m(en.)53 b(If)36 b Fn(n)f Fo(is)h(o)s(dd)f(then,)60 4077 y(b)s(ecause)g Fn(v)469 4092 y Fv(n)550 4077 y Fo(is)g(ev)m(en,)h(w)m(e)f(m)m(ust)g(ha)m(v)m(e) g(either)g Fn(P)47 b Fo(ev)m(en)35 b(and)f Fn(Q)h Fo(o)s(dd)e(or)h (\(from)f(the)i(recurrence\))h Fn(P)47 b Fo(and)60 4197 y Fn(Q)33 b Fo(b)s(oth)f(o)s(dd)g(and)h(3)28 b Fm(j)f Fn(n)p Fo(.)44 b(So)32 b(w)m(e)i(ha)m(v)m(e)g(either)f Fn(P)46 b Fo(ev)m(en)34 b(or)e Fn(Q)h Fo(o)s(dd)g(and)f(3)c Fm(j)f Fn(n)p Fo(.)206 4318 y(Con)m(v)m(ersely)-8 b(,)34 b(assume)e(that)e Fn(n)e Fm(2)g Fn(T)44 b Fo(is)31 b(a)f(pro)s(duct)g (of)g(sp)s(ecial)h(primes,)h(and)f(either)g Fn(P)43 b Fo(is)31 b(ev)m(en)h(or)e(\()p Fn(Q)60 4438 y Fo(is)k(o)s(dd)f(and)h(3) 29 b Fm(j)g Fn(n)p Fo(\).)46 b(W)-8 b(e)34 b(kno)m(w)g(from)f (Corollary)p 0 .5 0 TeXcolorrgb 1934 4438 a SDict begin H.S end 1934 4438 a 0 .5 0 TeXcolorrgb Fo(17)p 0 .5 0 TeXcolorrgb 2032 4374 a SDict begin H.R end 2032 4374 a 2032 4438 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.17) cvn H.B /ANN pdfmark end 2032 4438 a Black 33 w Fo(that)g(ev)m(ery)j(pro)s(duct)d(of)g(sp)s(ecial)i(primes)g(is)e(in)h Fn(T)14 b Fo(.)60 4558 y(So)33 b(if)g(2)g(is)h(sp)s(ecial,)h(then)f(2)p Fn(n)29 b Fm(2)g Fn(T)14 b Fo(.)45 b(So)33 b(w)m(e)i(can)e(assume)i(2)e (is)h(not)f(sp)s(ecial,)i(and)e(hence)i(that)e Fn(n)g Fo(is)h(o)s(dd.)60 4679 y(If)g Fn(P)48 b Fo(is)35 b(ev)m(en,)i(then,)f (from)e(the)h(recurrence,)i(all)d(the)h Fn(v)2131 4694 y Fv(k)2174 4679 y Fo(,)g(in)g(particular)f Fn(v)2850 4694 y Fv(n)2932 4679 y Fo(and)h Fn(v)3171 4694 y Ft(2)p Fv(n)3253 4679 y Fo(,)g(are)g(ev)m(en.)50 b(Also,)60 4799 y(if)37 b Fn(P)51 b Fo(and)38 b Fn(Q)f Fo(are)h(b)s(oth)f(o)s(dd)g (and)h(3)d Fm(j)h Fn(n)p Fo(,)j(then)f Fn(v)1918 4814 y Fv(n)2002 4799 y Fo(and)g Fn(v)2244 4814 y Ft(2)p Fv(n)2362 4799 y Fo(=)e Fn(v)2525 4763 y Ft(2)2521 4824 y Fv(n)2593 4799 y Fm(\000)26 b Fo(2)p Fn(Q)2822 4763 y Fv(n)2907 4799 y Fo(are)37 b(b)s(oth)g(ev)m(en.)60 b(Since)38 b(for)60 4920 y(ev)m(ery)c(prime)g(factor)e Fn(q)k Fo(of)d Fn(n)f Fo(with)i Fn(q)1424 4883 y Fv(\025)1469 4920 y Fm(k)p Fn(n)e Fo(w)m(e)i(ha)m(v)m(e)g Fn(\025)28 b Fm(\024)g Fo(log)2294 4943 y Fv(q)2348 4920 y Fn(n)g(<)g(n)p Fo(,)33 b(so)g(that)f Fn(n)c Fm(j)g Fn(Q)3206 4883 y Fv(n)3253 4920 y Fo(.)43 b(Hence)34 b(2)p Fn(n)28 b Fm(j)f Fn(v)3850 4935 y Ft(2)p Fv(n)3933 4920 y Fo(,)60 5040 y(2)p Fn(n)h Fm(2)g Fn(T)14 b Fo(.)p Black 1961 5289 a(10)p Black eop end end %%Page: 11 11 TeXDict begin HPSdict begin 11 10 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.11) cvn H.B /DEST pdfmark end 60 -300 a Black Black 206 -200 a Fo(The)34 b(pro)s(of)e(of)g(part)g(\(b\))h(is)g(just)g(the)g (same)g(as)g(that)f(for)g(Theorem)p 0 .5 0 TeXcolorrgb 2742 -200 a SDict begin H.S end 2742 -200 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 2791 -265 a SDict begin H.R end 2791 -265 a 2791 -200 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 2791 -200 a Black Fo(\(b\).)206 -80 y(T)-8 b(o)45 b(pro)m(v)m(e)i(part)d (\(c\):)69 b(w)m(e)46 b(see)g(easily)g(from)f(Theorem)p 0 .5 0 TeXcolorrgb 2350 -80 a SDict begin H.S end 2350 -80 a 0 .5 0 TeXcolorrgb Fo(15)p 0 .5 0 TeXcolorrgb 2448 -144 a SDict begin H.R end 2448 -144 a 2448 -80 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 2448 -80 a Black 45 w Fo(and)g(Prop)s(osition)p 0 .5 0 TeXcolorrgb 3232 -80 a SDict begin H.S end 3232 -80 a 0 .5 0 TeXcolorrgb Fo(18)p 0 .5 0 TeXcolorrgb 3329 -144 a SDict begin H.R end 3329 -144 a 3329 -80 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.18) cvn H.B /ANN pdfmark end 3329 -80 a Black 45 w Fo(that)g(the)g(only)60 40 y(p)s(ossible)36 b(\(second)g(kind\))f(basic)g(n)m(um)m(b)s(ers)i(are)d (1)g(and)h(6.)49 b(T)-8 b(o)34 b(\014nd)h(the)g(conditions)h(on)e Fn(P)48 b Fo(and)34 b Fn(Q)h Fo(that)60 161 y(mak)m(e)g(6)f(basic,)i(w) m(e)f(assume)h(that)e(6)c Fm(2)h Fn(T)48 b Fo(but)34 b(2)42 b Fn(=)-61 b Fm(2)31 b Fn(T)14 b Fo(,)34 b(3)42 b Fn(=)-60 b Fm(2)31 b Fn(T)14 b Fo(.)48 b(Then)35 b Fn(v)2765 176 y Ft(2)2835 161 y Fo(=)30 b Fn(P)3018 125 y Ft(2)3080 161 y Fm(\000)24 b Fo(2)p Fn(Q)34 b Fo(is)h(o)s(dd,)f(so)h Fn(P)47 b Fo(is)60 281 y(o)s(dd.)c(Also)32 b(3)c Fe(-)f Fn(v)688 296 y Ft(3)755 281 y Fo(=)h Fn(P)14 b Fo(\()p Fn(P)1051 245 y Ft(2)1109 281 y Fm(\000)21 b Fo(3)p Fn(Q)p Fo(\),)32 b(so)g Fn(P)41 b Fm(\021)28 b(\006)p Fo(1)45 b(\(mo)s(d)33 b(6\).)43 b(F)-8 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1072 y Fo(In)42 b(this)h(section)g(w)m(e)g (lo)s(ok)f(at)f(when)i Fn(S)6 b Fo(,)45 b Fm(P)1699 1087 y Fv(S)1750 1072 y Fo(,)f(and)e Fn(T)14 b Fo(,)44 b Fm(P)2231 1087 y Fv(T)2328 1072 y Fo(are)e(\014nite.)73 b(The)43 b(results)g(giv)m(en)g(here)g(are)60 1192 y(essen)m(tially)35 b(reform)m(ulations)f(of)e(results)i(of)e(Somer)h([)p 0 .5 0 TeXcolorrgb 2030 1192 a SDict begin H.S end 2030 1192 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 2128 1128 a SDict begin H.R end 2128 1128 a 2128 1192 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 2128 1192 a Black Fo(],)f([)p 0 .5 0 TeXcolorrgb 2241 1192 a SDict begin H.S end 2241 1192 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 2339 1128 a SDict begin H.R end 2339 1128 a 2339 1192 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1393479) cvn H.B /ANN pdfmark end 2339 1192 a Black Fo(].)60 1217 y SDict begin H.S end 60 1217 a 60 1217 a SDict begin 14.5 H.A end 60 1217 a 60 1217 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.22) cvn H.B /DEST pdfmark end 60 1217 a Black 167 x Fk(Theorem)50 b(22.)p Black 47 w Fl(The)43 b(set)h Fn(S)50 b Fl(is)44 b(\014nite)g(if)g(and)f (only)h(if)g Fo(\001)h(=)g(1)p Fl(,)h(in)e(which)f(c)-5 b(ase)44 b Fn(S)50 b Fo(=)45 b Fm(f)p Fo(1)p Fm(g)p Fl(.)72 b(F)-7 b(or)43 b Fn(S)60 1505 y Fl(in\014nite,)33 b Fm(P)494 1520 y Fv(S)578 1505 y Fl(is)f(\014nite)h(when)f Fn(Q)c Fo(=)g(0)k Fl(and)h Fn(P)41 b Fm(6)p Fo(=)28 b(0)p Fl(,)33 b(in)f(which)g(c)-5 b(ase)33 b Fm(P)2641 1520 y Fv(S)2725 1505 y Fl(c)-5 b(onsists)32 b(of)h(the)g(prime)f(divisors)60 1625 y(of)j Fn(P)14 b Fl(.)44 b(Otherwise,)34 b Fm(P)882 1640 y Fv(S)967 1625 y Fl(is)h(also)f(in\014nite.)44 b(F)-7 b(urthermor)i(e,)34 b Fm(P)2309 1640 y Fv(S)2395 1625 y Fl(is)g(the)h(set)g Fm(P)43 b Fl(of)35 b(al)5 b(l)34 b(primes)g(if)h(and)f(only)60 1745 y(if)h(every)f(prime)h (divisor)f(of)g Fn(Q)h Fl(is)g(sp)-5 b(e)g(cial.)44 b(\(This)34 b(includes)g(the)h(c)-5 b(ase)34 b Fn(Q)28 b Fo(=)f Fm(\006)p Fo(1)p Fl(.\))206 1937 y Fo(F)-8 b(or)32 b(the)h(pro)s(of,)f(w)m(e)i (note)e(\014rst)h(that)g(when)g(\001)28 b(=)g(1,)k Fn(\013)i Fo(and)e Fn(\014)38 b Fo(are)33 b(consecutiv)m(e)i(in)m(tegers,)f(and)f (1)f(is)60 2058 y(the)j(only)h(basic)f(elemen)m(t.)52 b(But)35 b(there)h(are)f(no)g(primes)h Fn(p)e Fo(dividing)i Fn(u)2684 2073 y Ft(1)2723 2058 y Fo(\001)c(=)g(1,)j(so)g Fm(P)3246 2073 y Fv(S;)p Ft(1)3383 2058 y Fo(is)g(empt)m(y)-8 b(,)37 b(and)60 2178 y Fn(S)k Fo(=)36 b Fm(f)p Fo(1)p Fm(g)p Fo(.)56 b(In)38 b(all)f(other)g(cases,)j Fm(j)p Fn(u)1393 2193 y Ft(1)1432 2178 y Fo(\001)p Fm(j)35 b Fn(>)h Fo(1,)i Fm(P)1871 2193 y Fv(S;)p Ft(1)2010 2178 y Fo(is)g(nonempt)m(y)-8 b(,)40 b(with)e Fn(p)d Fm(2)h(P)3075 2193 y Fv(S;)p Ft(1)3214 2178 y Fo(sa)m(y)-8 b(,)40 b(and)d(then,)i(b)m (y)60 2299 y(Corollary)p 0 .5 0 TeXcolorrgb 491 2299 a SDict begin H.S end 491 2299 a 0 .5 0 TeXcolorrgb Fo(2)p 0 .5 0 TeXcolorrgb 540 2234 a SDict begin H.R end 540 2234 a 540 2299 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.2) cvn H.B /ANN pdfmark end 540 2299 a Black Fo(,)32 b Fn(p)648 2262 y Fv(k)719 2299 y Fm(2)c Fn(S)38 b Fo(for)32 b(all)g Fn(k)f Fm(\025)d Fo(0,)33 b(making)g Fn(S)38 b Fo(in\014nite.)206 2419 y(No)m(w)d(assume)g Fn(S)k Fo(is)c(in\014nite.)47 b(W)-8 b(e)35 b(recall)f(that)f(\()p Fn(u)2075 2434 y Fv(n)2122 2419 y Fo(\))2160 2434 y Fv(n)p Fu(\025)p Ft(0)2331 2419 y Fo(is)h(called)g Fl(de)-5 b(gener)g(ate)33 b Fo(if)h Fn(Q)29 b Fo(=)h(0)j(or)h Fn(\013)q(=\014)k Fo(is)60 2539 y(a)f(ro)s(ot)f(of)g(unit)m(y)-8 b(.)58 b(\(The)37 b(latter)g(alternativ)m(e)h(includes)h(the)f(case)f Fn(P)49 b Fo(=)35 b(0,)j Fn(Q)d Fm(6)p Fo(=)g(0.\))56 b(W)-8 b(e)37 b(consider)i(the)60 2660 y(t)m(w)m(o)c(cases)h(of)e(\()p Fn(u)700 2675 y Fv(n)747 2660 y Fo(\))g(b)s(eing)h(degenerate)g(or)f (nondegenerate)i(separately)-8 b(.)51 b(If)34 b(\()p Fn(u)3034 2675 y Fv(n)3081 2660 y Fo(\))g(is)h(degenerate,)h(then)60 2780 y(b)m(y)d([)p 0 .5 0 TeXcolorrgb 222 2780 a SDict begin H.S end 222 2780 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 320 2716 a SDict begin H.R end 320 2716 a 320 2780 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 320 2780 a Black Fo(,)g(Theorem)h(9])e (either)p Black 205 2972 a Fm(\017)p Black 49 w Fn(P)41 b Fm(6)p Fo(=)28 b(0)j(and)g Fn(Q)d Fo(=)g(0,)j(so)h(that)f(then)h Fn(S)37 b Fo(consists)c(of)e(those)h Fn(n)g Fo(whose)h(prime)f(factors) f(all)h(divide)g Fn(P)14 b Fo(,)304 3092 y(and)33 b Fm(P)563 3107 y Fv(S)641 3092 y Fo(=)28 b Fm(P)814 3107 y Ft(1st)941 3092 y Fo(is)33 b(the)g(set)h(of)e(prime)h(divisors)h(of)e Fn(P)14 b Fo(;)304 3252 y(or)p Black 205 3452 a Fm(\017)p Black 49 w Fo(for)44 b(some)i Fn(r)51 b Fo(=)d(1)p Fn(;)17 b Fo(2)p Fn(;)g Fo(3)p Fn(;)g Fo(4)43 b(or)h(6,)k Fn(S)j Fo(has)45 b(a)f(subset)j Fm(f)p Fn(r)s(k)k Fo(:)d Fn(k)k Fm(2)c Fe(N)p Fm(g)d Fo(where)h Fn(u)3277 3467 y Fv(r)r(k)3401 3452 y Fo(=)j(0,)e(so)e(that)304 3572 y Fm(P)373 3587 y Fv(S)452 3572 y Fo(=)27 b Fm(P)624 3587 y Ft(1st)747 3572 y Fo(=)g Fm(P)8 b Fo(.)206 3765 y(No)m(w)36 b(consider)h(the)e (case)h(of)f(\()p Fn(u)1401 3780 y Fv(n)1447 3765 y Fo(\))g (nondegenerate.)53 b(Then,)37 b(b)m(y)f(Somer)g([)p 0 .5 0 TeXcolorrgb 2962 3765 a SDict begin H.S end 2962 3765 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 3060 3700 a SDict begin H.R end 3060 3700 a 3060 3765 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 3060 3765 a Black Fo(,)f(Theorem)i(1],)f(all)f(but)60 3885 y(\014nitely)g(man)m(y)f Fn(u)711 3900 y Fv(n)791 3885 y Fo(ha)m(v)m(e)h(a)e(primitiv)m(e)i(prime)g(divisor)f(\(a)f (prime)i(dividing)f Fn(u)2944 3900 y Fv(n)3024 3885 y Fo(that)f(do)h(not)f(divide)i Fn(u)3894 3900 y Fv(m)60 4005 y Fo(for)30 b(an)m(y)i Fn(m)c(<)f(n)p Fo(\).)43 b(So,)31 b(using)h(Theorem)p 0 .5 0 TeXcolorrgb 1597 4005 a SDict begin H.S end 1597 4005 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 1646 3941 a SDict begin H.R end 1646 3941 a 1646 4005 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 1646 4005 a Black Fo(\(a\),)e Fm(P)1897 4020 y Fv(S)1979 4005 y Fo(is)i(in\014nite.)43 b(Somer's)32 b(theorem)g(is)f(based)h(on)f(results)60 4126 y(of)e(Lekk)m(erk)m(erk)m(er)34 b([)p 0 .5 0 TeXcolorrgb 771 4126 a SDict begin H.S end 771 4126 a 0 .5 0 TeXcolorrgb Fo(14)p 0 .5 0 TeXcolorrgb 869 4061 a SDict begin H.R end 869 4061 a 869 4126 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0055373) cvn H.B /ANN pdfmark end 869 4126 a Black Fo(])29 b(and)h(Sc)m(hinzel)i([)p 0 .5 0 TeXcolorrgb 1513 4126 a SDict begin H.S end 1513 4126 a 0 .5 0 TeXcolorrgb Fo(18)p 0 .5 0 TeXcolorrgb 1611 4061 a SDict begin H.R end 1611 4061 a 1611 4126 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0139567) cvn H.B /ANN pdfmark end 1611 4126 a Black Fo(].)42 b(In)30 b(fact)g(Bilu,)h(Hanrot)e(and)h(V)-8 b(outier)30 b([)p 0 .5 0 TeXcolorrgb 3139 4126 a SDict begin H.S end 3139 4126 a 0 .5 0 TeXcolorrgb Fo(5)p 0 .5 0 TeXcolorrgb 3188 4061 a SDict begin H.R end 3188 4061 a 3188 4126 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1863855) cvn H.B /ANN pdfmark end 3188 4126 a Black Fo(])g(ha)m(v)m(e)h (pro)m(v)m(ed)g(that)60 4246 y(for)26 b(suc)m(h)i(sequences)i(with)e (no)e(sp)s(ecial)i(primes)g(ev)m(ery)h Fn(u)2129 4261 y Fv(n)2202 4246 y Fo(with)e Fn(n)h(>)g Fo(30)e(has)h(a)f(primitiv)m(e) j(divisor.)43 b(They)60 4366 y(also)27 b(listed)h(exceptions)h(with)f Fn(n)g Fm(\024)g Fo(30.)41 b(Hence)28 b Fn(u)1888 4386 y Fv(p)1924 4367 y Fi(k)1992 4366 y Fo(has)g(a)f(primitiv)m(e)h(prime)g (divisor)g(for)e(all)h(su\016cien)m(tly)60 4487 y(large)33 b Fn(k)s Fo(,)g(making)g Fm(P)826 4502 y Fv(S)910 4487 y Fo(in\014nite.)46 b(See)34 b(Ab)s(ouzaid)f([)p 0 .5 0 TeXcolorrgb 1920 4487 a SDict begin H.S end 1920 4487 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 1969 4423 a SDict begin H.R end 1969 4423 a 1969 4487 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR2289425) cvn H.B /ANN pdfmark end 1969 4487 a Black Fo(])g(for)f(corrections)i(to)f(their)g(list.) 45 b(Also)34 b(Stew)m(art)f([)p 0 .5 0 TeXcolorrgb 3835 4487 a SDict begin H.S end 3835 4487 a 0 .5 0 TeXcolorrgb Fo(22)p 0 .5 0 TeXcolorrgb 3933 4423 a SDict begin H.R end 3933 4423 a 3933 4487 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR0491445) cvn H.B /ANN pdfmark end 3933 4487 a Black Fo(])60 4607 y(and)g(Shorey)g(and)g(Stew)m(art)g([)p 0 .5 0 TeXcolorrgb 1149 4607 a SDict begin H.S end 1149 4607 a 0 .5 0 TeXcolorrgb Fo(19)p 0 .5 0 TeXcolorrgb 1247 4543 a SDict begin H.R end 1247 4543 a 1247 4607 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR602235) cvn H.B /ANN pdfmark end 1247 4607 a Black Fo(])f(ga)m(v)m(e)i(lo)m(w)m(er)g(b)s(ounds)f (for)f(the)h(largest)g(prime)g(divisor)h(of)e Fn(u)3513 4622 y Fv(n)3559 4607 y Fo(.)206 4728 y(This)47 b(pro)s(of)d(will)i(b)s (e)f(complete)i(after)e(w)m(e)h(ha)m(v)m(e)g(pro)m(v)m(ed)h(the)f (follo)m(wing.)81 b(While)46 b(this)g(result)g(is)60 4848 y(con)m(tained)34 b(in)e(Somer)i([)p 0 .5 0 TeXcolorrgb 941 4848 a SDict begin H.S end 941 4848 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 1038 4784 a SDict begin H.R end 1038 4784 a 1038 4848 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 1038 4848 a Black Fo(,)f(Theorem)h(8],)e(w)m(e)i(giv)m(e)f(another)g(pro)s (of)f(here.)60 4873 y SDict begin H.S end 60 4873 a 60 4873 a SDict begin 14.5 H.A end 60 4873 a 60 4873 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.23) cvn H.B /DEST pdfmark end 60 4873 a Black 167 x Fk(Prop)s(osition)38 b(23.)p Black 41 w Fl(The)d(set)f Fm(P)1270 5055 y Fv(S)1356 5040 y Fl(is)h(the)g(whole)f(of)g Fm(P)44 b Fl(if)34 b(and)h(only)f(if)h(al)5 b(l)35 b(primes)f(ar)-5 b(e)34 b(r)-5 b(e)g(gular.)p Black 1961 5289 a Fo(11)p Black eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end 60 -300 a Black Black Black 100 x Fl(Pr)-5 b(o)g(of.)p Black 48 w Fo(First)40 b(note)h(that)f(if)f (there)i(are)f(an)m(y)h(irregular)f(primes)h(then,)i(b)m(y)e(Corollary) p 0 .5 0 TeXcolorrgb 3334 -200 a SDict begin H.S end 3334 -200 a 0 .5 0 TeXcolorrgb Fo(7)p 0 .5 0 TeXcolorrgb 3383 -265 a SDict begin H.R end 3383 -265 a 3383 -200 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 3383 -200 a Black Fo(,)h Fm(P)3521 -185 y Fv(S)3572 -200 y Fo(,)g(b)s(eing)e(a)60 -80 y(subset)34 b(of)e Fm(P)539 -65 y Ft(1st)634 -80 y Fo(,)h(cannot)g(b)s(e)f(the)h(whole)h (of)e Fm(P)8 b Fo(.)206 40 y(Con)m(v)m(ersely)-8 b(,)42 b(assume)c(all)f(primes)h(are)f(regular,)h(so)g(that)e(an)m(y)i(prime)g (factor)e Fn(p)h Fo(of)g Fn(Q)g Fo(also)g(divides)60 161 y Fn(P)14 b Fo(.)52 b(Note)36 b(that)g(then)g Fn(p)d Fm(j)g Fo(\001.)53 b(T)-8 b(o)36 b(sho)m(w)h(that)e(all)h(primes)h(b)s (elong)f(to)f Fm(P)2768 176 y Fv(S)2819 161 y Fo(,)i(w)m(e)g(pro)s (ceed)f(b)m(y)h(induction.)60 281 y(W)-8 b(e)36 b(\014rst)f(sho)m(w)i (that)e(2)d Fm(2)g(P)1141 296 y Fv(S)1192 281 y Fo(.)52 b(If)35 b Fn(u)1427 296 y Ft(2)1498 281 y Fo(=)d Fn(P)48 b Fo(is)36 b(ev)m(en,)i(then)e(2)31 b Fm(2)i Fn(S)6 b Fo(,)36 b(2)c Fm(2)g(P)2852 296 y 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Fm(j)h Fn(u)3408 854 y Ft(6)3447 839 y Fo(.)p Black 205 1040 a Fm(\017)p Black 49 w Fo(If)k Fn(P)41 b Fm(\021)29 b Fo(3)44 b(\(mo)s(d)33 b(6\),)f Fn(Q)c Fm(\021)g(\006)p Fo(1)45 b(\(mo)s(d)33 b(6\),)f(then)h(6)g(is)g(basic,)g(b)m(y)h (Theorem)p 0 .5 0 TeXcolorrgb 3114 1040 a SDict begin H.S end 3114 1040 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 3162 976 a SDict begin H.R end 3162 976 a 3162 1040 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 3162 1040 a Black Fo(\(c\).)p Black 205 1241 a Fm(\017)p Black 49 w Fo(If)e Fn(P)41 b Fm(\021)29 b(\006)p Fo(1)44 b(\(mo)s(d)33 b(6\),)g Fn(Q)28 b Fm(\021)g(\000)p Fo(1)44 b(\(mo)s(d)33 b(6\),)g(then)g(12)f (is)h(basic,)h(b)m(y)f(Theorem)p 0 .5 0 TeXcolorrgb 3240 1241 a SDict begin H.S end 3240 1241 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 3289 1177 a SDict begin H.R end 3289 1177 a 3289 1241 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.1) cvn H.B /ANN pdfmark end 3289 1241 a Black Fo(\(c\).)p Black 205 1443 a Fm(\017)p Black 49 w Fo(If)f Fn(P)41 b Fm(\021)29 b(\006)p Fo(1)44 b(\(mo)s(d)33 b(6\),)g Fn(Q)28 b Fm(\021)g Fo(1)44 b(\(mo)s(d)33 b(6\),)f(then)h(3)28 b Fm(j)f Fn(u)2303 1458 y Ft(3)2375 1443 y Fo(and)32 b(so)h(3)28 b Fm(j)f Fn(u)2872 1458 y Ft(3)2911 1443 y Fn(v)2958 1458 y Ft(3)3025 1443 y Fo(=)h Fn(u)3185 1458 y Ft(6)3224 1443 y Fo(.)60 1639 y(Hence)34 b(2)27 b Fm(2)h(P)589 1654 y Fv(S)641 1639 y Fo(,)k(as)h(claimed.)206 1760 y(W)-8 b(e)38 b(no)m(w)g(assume)h(that)e Fn(q)i Fm(2)d(P)1398 1775 y Fv(S)1487 1760 y Fo(for)g(ev)m(ery)j(prime)g Fn(q)g(<)d(p)p Fo(,)i(where)g Fn(p)g Fo(is)f(a)g(prime)h(at)f(least)h (3.)58 b(W)-8 b(e)60 1880 y(ha)m(v)m(e)36 b(just)g(sho)m(wn)g(that)f (this)g(is)h(true)f(for)f Fn(p)e Fo(=)g(3.)50 b(By)35 b(Lemma)p 0 .5 0 TeXcolorrgb 2468 1880 a SDict begin H.S end 2468 1880 a 0 .5 0 TeXcolorrgb Fo(8)p 0 .5 0 TeXcolorrgb 2517 1816 a SDict begin H.R end 2517 1816 a 2517 1880 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 2517 1880 a Black Fo(,)g(w)m(e)h(kno)m(w)g (that)f(for)f(an)m(y)i(exp)s(onen)m(ts)60 2000 y Fn(")106 2015 y Fv(q)183 2000 y Fo(=)k(0)f(or)g(1)h(there)g(is)g(a)f(p)s(ositiv) m(e)i(in)m(teger)g Fn(k)h Fo(suc)m(h)f(that)f Fn(k)2358 1926 y Ff(Q)2452 2029 y Fv(q)r(
)f Fo(2,)j(all)g(prime)g(factors)g(of)f Fn(p)16 b Fo(+)g Fn(")29 b Fo(are)h(less)h(than)60 2362 y Fn(p)37 b Fo(so,)i(b)m(y)f(the)g(induction)g(h)m(yp)s(othesis,)j Fn(k)s Fo(\()p Fn(p)25 b Fo(+)g Fn(")p Fo(\))35 b Fm(2)h Fn(S)43 b Fo(for)37 b(some)h Fn(k)s Fo(.)58 b(If)37 b Fn(p)e Fm(j)h Fn(k)k Fo(then)e Fn(p)d Fm(2)h(P)3520 2377 y Fv(S)3571 2362 y Fo(.)58 b(If)37 b Fn(p)f Fe(-)f Fn(k)60 2482 y Fo(then,)e(using)h(\()p 0 .5 0 TeXcolorrgb 603 2482 a SDict begin H.S end 603 2482 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 651 2418 a SDict begin H.R end 651 2418 a 651 2482 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (equation.1) cvn H.B /ANN pdfmark end 651 2482 a Black Fo(\),)f(w)m(e)g(ha)m (v)m(e)1196 2708 y Fn(u)1252 2723 y Fv(pk)r Ft(\()p Fv(p)p Ft(+)p Fv(")p Ft(\))1535 2708 y Fo(=)28 b Fn(u)1695 2667 y Ft(\()p Fv(k)r Ft(\()p Fv(p)p Ft(+)p Fv(")p Ft(\)\))1695 2732 y Fv(p)1970 2708 y Fn(u)2026 2723 y Fv(k)r Ft(\()p Fv(p)p Ft(+)p Fv(")p Ft(\))2273 2708 y Fo(=)g Fn(u)2433 2657 y Ft(\()p Fv(p)p Ft(+)p Fv(")p Ft(\))2433 2735 y Fv(pk)2614 2708 y Fn(u)2670 2723 y Fv(p)p Ft(+)p Fv(")2797 2708 y Fn(;)60 2920 y Fo(so)33 b(that)f Fn(pk)s Fo(\()p Fn(p)22 b Fo(+)g Fn(")p Fo(\))27 b Fm(2)i Fn(S)6 b Fo(,)32 b Fn(p)c Fm(2)g(P)1272 2935 y Fv(S)1323 2920 y Fo(.)43 b(This)34 b(pro)m(v)m(es)g(the)f(induction)h(step.)p 3889 2920 4 66 v 3893 2857 59 4 v 3893 2920 V 3951 2920 4 66 v 206 3113 a(This)g(completes)g(the)f(pro)s(of)f(of)g(Theorem)p 0 .5 0 TeXcolorrgb 1825 3113 a SDict begin H.S end 1825 3113 a 0 .5 0 TeXcolorrgb Fo(22)p 0 .5 0 TeXcolorrgb 1923 3049 a SDict begin H.R end 1923 3049 a 1923 3113 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.22) cvn H.B /ANN pdfmark end 1923 3113 a Black Fo(.)206 3233 y(W)-8 b(e)33 b(no)m(w)h(consider)f(the)g(\014niteness)i(\(or)d(otherwise\))i(of)f Fn(T)46 b Fo(and)32 b Fm(P)2651 3248 y Fv(T)2707 3233 y Fo(.)60 3258 y SDict begin H.S end 60 3258 a 60 3258 a SDict begin 14.5 H.A end 60 3258 a 60 3258 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.24) cvn H.B /DEST pdfmark end 60 3258 a Black 172 x Fk(Theorem)g(24)f Fo(\(Somer)d([)p 0 .5 0 TeXcolorrgb 1030 3430 a SDict begin H.S end 1030 3430 a 0 .5 0 TeXcolorrgb Fo(21)p 0 .5 0 TeXcolorrgb 1128 3366 a SDict begin H.R end 1128 3366 a 1128 3430 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1393479) cvn H.B /ANN pdfmark end 1128 3430 a Black Fo(,)g(Theorems)h(8)e(and)g(9]\))p Fk(.)p Black 37 w Fl(The)j(set)g Fn(T)44 b Fl(is)29 b(\014nite)h(in)f (the)h(fol)5 b(lowing)29 b(two)h(c)-5 b(ases:)p Black 205 3627 a Fm(\017)p Black 49 w Fn(P)41 b Fo(=)28 b Fm(\006)p Fo(1)p Fl(,)35 b Fn(Q)27 b Fm(6\021)i(\000)p Fo(1)44 b(\(mo)s(d)33 b(6\))p Fl(,)i(in)f(which)g(c)-5 b(ase)35 b Fn(T)41 b Fo(=)27 b Fm(f)p Fo(1)p Fm(g)p Fl(;)p Black 205 3828 a Fm(\017)p Black 49 w Fn(P)41 b Fo(=)28 b Fn(")558 3843 y Ft(1)597 3828 y Fo(2)646 3792 y Fv(k)688 3828 y Fl(,)33 b Fn(Q)28 b Fo(=)g(2)1009 3792 y Ft(2)p Fv(k)r Fu(\000)p Ft(1)1194 3828 y Fo(+)17 b Fn(")1333 3843 y Ft(2)1372 3828 y Fl(,)33 b(wher)-5 b(e)32 b Fn(k)k Fl(is)d(a)f(p)-5 b(ositive)32 b(inte)-5 b(ger,)33 b(and)f Fn(")2913 3843 y Ft(1)2952 3828 y Fl(,)h Fn(")3061 3843 y Ft(2)3128 3828 y Fm(2)28 b(f\000)p Fo(1)p Fn(;)17 b Fo(1)p Fm(g)p Fl(,)32 b(in)h(which)304 3948 y(c)-5 b(ase)34 b Fn(T)41 b Fo(=)28 b Fm(f)p Fo(1)p Fn(;)17 b Fo(2)p Fm(g)p Fl(.)60 4145 y(Otherwise,)35 b Fn(T)49 b Fl(is)35 b(in\014nite.)46 b(F)-7 b(or)35 b Fn(T)49 b Fl(in\014nite,)35 b Fm(P)1862 4160 y Fv(T)1953 4145 y Fl(is)g(\014nite)g(pr)-5 b(e)g(cisely)35 b(when)g Fn(P)s(;)17 b(Q)35 b Fl(ar)-5 b(e)35 b(not)h(b)-5 b(oth)35 b Fo(0)h Fl(and)60 4265 y(either)p Black 205 4462 a Fm(\017)p Black 49 w Fn(P)381 4426 y Ft(2)447 4462 y Fo(=)28 b Fn(Q)p Fl(,)35 b(in)g(which)f(c)-5 b(ase)34 b Fm(P)1365 4477 y Fv(T)1455 4462 y Fl(is)h(the)f(set)h(of)g(prime)f (divisors)g(of)h Fo(2)p Fn(P)304 4623 y Fl(or)p Black 205 4824 a Fm(\017)p Black 49 w Fn(P)381 4788 y Ft(2)447 4824 y Fo(=)28 b(4)p Fn(Q)35 b Fl(or)g Fn(Q)28 b Fo(=)f(0)p Fl(,)35 b(in)f(which)g(c)-5 b(ase)34 b Fm(P)1831 4839 y Fv(T)1922 4824 y Fl(is)g(the)h(set)g(of)g(prime)f(divisors)g(of)g Fn(P)14 b Fl(.)60 5021 y(Otherwise,)34 b(for)h Fn(T)48 b Fl(in\014nite,)34 b Fm(P)1243 5036 y Fv(T)1334 5021 y Fl(is)g(also)g(in\014nite.)p Black 1961 5289 a Fo(12)p Black eop end end %%Page: 13 13 TeXDict begin HPSdict begin 13 12 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.13) cvn H.B /DEST pdfmark end 60 -300 a Black Black Black 100 x Fl(Pr)-5 b(o)g(of.)p Black 48 w Fo(If)42 b Fn(T)55 b 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Fn(v)2741 898 y Ft(2)2815 883 y Fo(has)g(an)g(o)s(dd)g(prime)h(factor)e Fn(p)p Fo(,)60 1003 y(then)g(2)p Fn(p)381 967 y Fv(k)453 1003 y Fm(2)c Fn(T)47 b Fo(for)33 b(all)g Fn(k)g Fm(\025)d Fo(0,)j(so)h(that)f Fn(T)48 b Fo(is)34 b(again)f(in\014nite.)47 b(Finally)-8 b(,)34 b(if)g Fn(v)2898 1018 y Ft(2)2971 1003 y Fo(is)g Fm(\006)g Fo(a)f(p)s(o)m(w)m(er)i(of)e(2,)g(then)60 1124 y Fn(T)41 b Fo(=)28 b Fm(f)p Fo(1)p Fn(;)17 b Fo(2)p Fm(g)p Fo(.)41 b(This)31 b(happ)s(ens)f(only)g(when)g Fn(v)1680 1139 y Ft(2)1748 1124 y Fo(=)d(2)1900 1088 y Ft(2)p Fv(k)1993 1124 y Fm(\000)16 b Fo(2)p Fn(Q)28 b Fo(=)g Fm(\006)p Fo(2,)i(so)f(that)h Fn(Q)d Fo(=)h(2)3109 1088 y Ft(2)p Fv(k)r Fu(\000)p Ft(1)3292 1124 y Fm(\007)16 b Fo(1,)30 b(as)g(claimed.)206 1244 y(No)m(w)f(tak)m(e)g Fn(T)42 b Fo(in\014nite,)31 b(with)e Fn(P)s(;)17 b(Q)28 b Fo(not)g(b)s(oth)g(0.)42 b(If)28 b(the)h(sequence)i(\()p Fn(v)2773 1259 y Fv(n)2820 1244 y Fo(\))d(is)h(degenerate,)h(then,)g (using)60 1365 y(Somer)35 b([)p 0 .5 0 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3889 1967 4 66 v 3893 1904 59 4 v 3893 1967 V 3951 1967 4 66 v 60 2136 a SDict begin H.S end 60 2136 a 60 2136 a SDict begin 14.5 H.A end 60 2136 a 60 2136 a SDict begin [ /View [/XYZ H.V] /Dest (section.8) cvn H.B /DEST pdfmark end 60 2136 a 163 x Fp(8)161 b(The)53 b(sets)g Fb(P)1049 2321 y Fn(S)1172 2299 y Fp(and)h Fb(P)1584 2321 y Fn(T)1659 2299 y Fp(.)60 2518 y Fo(F)-8 b(rom)30 b(the)g(pro)s(of)f(of)h(Theorem) p 0 .5 0 TeXcolorrgb 1249 2518 a SDict begin H.S end 1249 2518 a 0 .5 0 TeXcolorrgb Fo(22)p 0 .5 0 TeXcolorrgb 1347 2454 a SDict begin H.R end 1347 2454 a 1347 2518 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.22) cvn H.B /ANN pdfmark end 1347 2518 a Black 30 w Fo(w)m(e)h(see)g(that)f Fm(P)1951 2533 y Fv(S)2030 2518 y Fo(=)d Fm(P)2202 2533 y Ft(1st)2327 2518 y Fo(for)j(\()p Fn(u)2568 2533 y Fv(n)2614 2518 y Fo(\))g(degenerate)i(or)d(all)h(primes)i(b)s(eing)60 2639 y(regular.)64 b(Our)39 b(next)h(result)g(tak)m(es)h(care)e(of)g (the)h(remaining)g(cases.)65 b(I)39 b(thank)h(Larry)f(Somer)h(and)g (the)60 2759 y(referee)34 b(for)e(p)s(oin)m(ting)g(out)h(ho)m(w)g(the)g (pro)s(of)f(of)g(this)h(could)g(b)s(e)g(completed.)60 2778 y SDict begin H.S end 60 2778 a 60 2778 a SDict begin 14.5 H.A end 60 2778 a 60 2778 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.25) cvn H.B /DEST pdfmark end 60 2778 a Black 184 x Fk(Prop)s(osition)28 b(25.)p Black 35 w Fl(If)e Fo(\()p Fn(u)1022 2977 y Fv(n)1069 2962 y Fo(\))h Fl(is)g(nonde)-5 b(gener)g(ate)25 b(and)i(ther)-5 b(e)27 b(ar)-5 b(e)26 b(irr)-5 b(e)g(gular)27 b(primes,)h(then)f Fm(P)3442 2977 y Fv(S)3520 2962 y Fl(is)g(a)g(pr)-5 b(op)g(er)60 3083 y(subset)35 b(of)f Fm(P)532 3098 y Ft(1st)628 3083 y Fl(.)p Black 60 3286 a(Pr)-5 b(o)g(of.)p Black 48 w Fo(T)d(ak)m(e)30 b(\()p Fn(u)691 3301 y Fv(n)738 3286 y Fo(\))e(nondegenerate)j(and)e(ha)m(ving)g(an)g(irregular)g(prime)g Fn(f)11 b Fo(.)42 b(Then,)31 b(from)d(the)i(discussion)60 3407 y(preceding)46 b(Prop)s(osition)p 0 .5 0 TeXcolorrgb 1048 3407 a SDict begin H.S end 1048 3407 a 0 .5 0 TeXcolorrgb Fo(23)p 0 .5 0 TeXcolorrgb 1145 3342 a SDict begin H.R end 1145 3342 a 1145 3407 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.23) cvn H.B /ANN pdfmark end 1145 3407 a Black Fo(,)i(ev)m(ery)e Fn(u)1545 3422 y Fv(n)1636 3407 y Fo(for)e Fn(n)h Fo(su\016cien)m(tly)i(large)e(has)g(a)f (primitiv)m(e)i(prime)g(divisor.)60 3527 y(Indeed,)e(if)c(gcd)q(\()p Fn(P)s(;)17 b(Q)p Fo(\))40 b(=)h(1)g(this)g(is)g(true)g(for)f Fn(n)h(>)g Fo(30.)67 b(Hence)42 b(for)e Fn(`)g Fo(su\016cien)m(tly)k (large,)e Fn(u)3615 3542 y Fv(`f)3729 3527 y Fo(has)f(a)60 3647 y(primitiv)m(e)34 b(prime)g(divisor,)f Fn(p)g Fo(sa)m(y)-8 b(,)33 b(so)g(that)f Fn(!)t Fo(\()p Fn(p)p Fo(\))27 b(=)h Fn(`f)11 b Fo(.)206 3768 y(Then)37 b(if,)g(for)e(some)i Fn(k)s Fo(,)g Fn(k)s(p)f Fo(w)m(ere)h(in)f Fn(S)6 b Fo(,)37 b(w)m(e)g(w)m(ould)g(ha)m(v)m(e)g Fn(k)s(p)c Fm(j)g Fn(u)2633 3783 y Fv(k)r(p)2711 3768 y Fo(,)k(so)f(that,)g(b)m(y)h([)p 0 .5 0 TeXcolorrgb 3306 3768 a SDict begin H.S end 3306 3768 a 0 .5 0 TeXcolorrgb Fo(20)p 0 .5 0 TeXcolorrgb 3404 3704 a SDict begin H.R end 3404 3704 a 3404 3768 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1271392) cvn H.B /ANN pdfmark end 3404 3768 a Black Fo(,)g(Prop)s(osition)60 3888 y(1\(iv\)],)28 b Fn(!)t Fo(\()p Fn(p)p Fo(\),)f(and)g(hence)h Fn(f)11 b Fo(,)27 b(w)m(ould)h(divide)g Fn(k)s(p)p Fo(.)41 b(Hence)28 b Fn(f)37 b Fo(w)m(ould)28 b(divide)g Fn(u)2873 3903 y Fv(n)2919 3888 y Fo(,)g(con)m(tradicting)g(Corollary)p 0 .5 0 TeXcolorrgb 60 4009 a SDict begin H.S end 60 4009 a 0 .5 0 TeXcolorrgb Fo(7)p 0 .5 0 TeXcolorrgb 109 3944 a SDict begin H.R end 109 3944 a 109 4009 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 109 4009 a Black Fo(.)43 b(Th)m(us)34 b Fn(p)28 b Fm(62)g(P)666 4024 y Fv(S)717 4009 y Fo(.)p 3889 4009 4 66 v 3893 3946 59 4 v 3893 4009 V 3951 4009 4 66 v 206 4204 a(W)-8 b(e)44 b(ha)m(v)m(e)h(in)f(fact)f(sho)m(wn)i(that)e(no)h(prime)g(whose)h(rank) f(of)f(app)s(earance)h(is)g(a)f(m)m(ultiple)i(of)e(an)m(y)60 4324 y(irregular)36 b(prime)h Fn(f)47 b Fo(will)37 b(b)s(elong)f(to)g Fm(P)1529 4339 y Fv(S)1580 4324 y Fo(.)55 b(The)37 b(referee)g(has)g (remark)m(ed)h(that,)f(when)g Fn(\013)q(=\014)42 b Fo(is)36 b(rational,)60 4444 y(the)e(densit)m(y)i(of)e(suc)m(h)h(primes)g(has)f (b)s(een)h(precisely)h(computed)f(in)f(man)m(y)h(cases.)49 b(F)-8 b(or)33 b Fn(f)41 b(>)29 b Fo(2)34 b(and)g Fn(\013)q(=\014)60 4565 y Fo(not)e(an)h Fn(f)11 b Fo(-th)32 b(p)s(o)m(w)m(er,)h(it)g(is)g Fn(f)11 b(=)p Fo(\()p Fn(f)1297 4529 y Ft(2)1358 4565 y Fm(\000)22 b Fo(1\).)44 b(See)33 b(Ballot)f([)p 0 .5 0 TeXcolorrgb 2106 4565 a SDict begin H.S end 2106 4565 a 0 .5 0 TeXcolorrgb Fo(4)p 0 .5 0 TeXcolorrgb 2155 4501 a SDict begin H.R end 2155 4501 a 2155 4565 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1257079) cvn H.B /ANN pdfmark end 2155 4565 a Black Fo(,)h(Theorem)h(3.2.3])e(and)g(also)h(Moree)g ([)p 0 .5 0 TeXcolorrgb 3594 4565 a SDict begin H.S end 3594 4565 a 0 .5 0 TeXcolorrgb Fo(16)p 0 .5 0 TeXcolorrgb 3692 4501 a SDict begin H.R end 3692 4501 a 3692 4565 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR2274151) cvn H.B /ANN pdfmark end 3692 4565 a Black Fo(].)206 4685 y(Using)h(a)e (similar)h(metho)s(d,)g(w)m(e)h(can)f(also)f(pro)m(v)m(e)i(the)f (corresp)s(onding)h(result)f(for)f Fn(T)14 b Fo(.)60 4705 y SDict begin H.S end 60 4705 a 60 4705 a SDict begin 14.5 H.A end 60 4705 a 60 4705 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.26) cvn H.B /DEST pdfmark end 60 4705 a Black 184 x Fk(Prop)s(osition)38 b(26.)p Black 41 w Fl(The)d(set)f Fm(P)1270 4904 y Fv(T)1361 4889 y Fl(is)g(a)h(pr)-5 b(op)g(er)34 b(subset)h(of)g Fm(P)2324 4904 y Ft(2nd)2442 4889 y Fl(.)p Black 1961 5289 a Fo(13)p Black eop end end %%Page: 14 14 TeXDict begin HPSdict begin 14 13 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 60 -300 a SDict begin H.S end 60 -300 a 0 TeXcolorgray 0 TeXcolorgray 60 -300 a SDict begin H.R end 60 -300 a 60 -300 a SDict begin [ /View [/XYZ H.V] /Dest (page.14) cvn H.B /DEST pdfmark end 60 -300 a Black Black Black 100 x Fl(Pr)-5 b(o)g(of.)p Black 48 w Fo(Let)44 b Fn(f)54 b Fo(b)s(e)44 b(a)f(primitiv)m(e)i(prime)f(divisor)h(of)e Fn(u)2111 -185 y Fv(n)2201 -200 y Fo(for)g(some)h(o)s(dd)f Fn(n)h Fo(with)g Fn(f)57 b Fe(-)46 b Fo(2)p Fn(Q)p Fo(.)76 b(Then,)47 b(b)m(y)60 -80 y(Prop)s(osition)p 0 .5 0 TeXcolorrgb 587 -80 a SDict begin H.S end 587 -80 a 0 .5 0 TeXcolorrgb Fo(19)p 0 .5 0 TeXcolorrgb 685 -144 a SDict begin H.R end 685 -144 a 685 -80 a SDict begin [ /Color [1 0 0] /H /I 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/Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.MR1393479) cvn H.B /ANN pdfmark end 1936 161 a Black Fo(,)h(Prop)s(osition)e (2\(vii\)],)i Fn(k)s(p)f Fo(is)f(a)g(m)m(ultiple)i(of)e Fn(`f)11 b Fo(.)58 b(In)60 281 y(particular,)33 b Fn(f)38 b Fm(j)28 b Fn(v)726 296 y Fv(k)r(p)804 281 y Fo(,)33 b(con)m(tradicting)g Fn(f)38 b Fm(62)29 b(P)1707 296 y Ft(2nd)1825 281 y Fo(.)43 b(So)33 b Fn(p)27 b Fm(62)h(P)2270 296 y Fv(T)2326 281 y Fo(.)p 3889 281 4 66 v 3893 219 59 4 v 3893 281 V 3951 281 4 66 v 60 454 a SDict begin H.S end 60 454 a 60 454 a SDict begin 14.5 H.A end 60 454 a 60 454 a SDict begin [ /View [/XYZ H.V] /Dest (section.9) cvn H.B /DEST pdfmark end 60 454 a 155 x Fp(9)161 b(Divisibilit)l(y)52 b(prop)t(erties)f(of)j Fg(S)63 b Fp(and)53 b(of)h Fg(T)20 b Fp(.)60 828 y Fo(F)-8 b(rom)30 b(Theorem)p 0 .5 0 TeXcolorrgb 724 828 a SDict begin H.S end 724 828 a 0 .5 0 TeXcolorrgb Fo(1)p 0 .5 0 TeXcolorrgb 773 764 a SDict begin H.R end 773 764 a 773 828 a SDict begin [ /Color 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