Journal of Integer Sequences, Vol. 1 (1998), Article 98.1.1 |

Department of Mathematics

Princeton University, Princeton NJ 08544

Email address: conway@math.Princeton.EDU

**Abstract:**(Supplied by the editors.) It is asserted without proof that every positive integer is the product of a unique "happy couple" of integers. A "happy couple" is an ordered pair of integers of one of three types: (*A*,*A*) ; (*B*,*C*), with*C*> 1, where there exist integers*R*,*S*such that*B**R*^{ 2}+ 1 =*C**S*^{ 2}; and (*D*,*E*) where there exist odd integers*T*,*U*such that*D**T*^{ 2}+ 2 =*E**U*^{ 2}.

Any ordered couple of integers that can be obtained
from the positive integers (*n*,*n*+*d*) by dividing them by
possibly distinct perfect squares prime to *d*
is called a "*d*-happy couple", except that couples of the form
(*m*,1) are NOT called 1-happy.

A "happy couple" is just a *d*-happy couple for *d* = 0, 1 or 2.

**Theorem.** *Each positive integer N is the product of a unique
happy couple.*

I call this "the happy factorization" of *N*, and append a table
of happy factorizations, writing a number as

according as it is the product of a

13^2 12^2 1.170 11^2 1.145 1:171 10^2 1.122 2.73 43.4 9^2 1.101 1:123 3.49 1.173 8^2 1.82 2.51 31.4 4.37 29.6 7^2 1.65 1:83 103:1 1.125 1.149 7.25 6^2 1.50 2.33 3.28 2:52 14.9 6.25 22:8 5^2 1.37 1:51 1:67 1.85 5.21 127:1 151:1 59.3 4^2 1.26 2.19 4.13 4.17 2.43 1.106 64:2 4:38 2.89 3^2 1.17 1:27 3.13 1.53 23.3 3:29 1:107 3.43 17.9 1:179 2^2 1.10 2.9 7.4 2:20 2.27 5.14 4:22 27.4 1.130 7.22 20.9 1^2 1.5 1:11 1:19 1.29 1.41 11.5 71:1 1.89 1.109 1:131 31.5 1.181 0^2 1.2 2.3 3.4 4.5 5.6 6.7 7.8 8.9 9.10 10.11 11.12 12.13 13.14 1^2 1:3 7:1 1.13 3.7 31:1 1:43 3.19 1.73 13:7 3.37 19.7 1.157 3.61 2^2 2:4 7.2 2.11 16:2 11.4 1.58 1.74 23.4 7.16 2.67 79.2 23.8 3^2 3:5 23:1 11.3 5.9 1:59 25:3 3.31 1.113 27:5 3.53 1.185 4^2 4:6 17.2 23.2 15.4 19.4 47.2 2.57 34:4 80:2 6.31 5^2 5:7 47:1 1.61 7.11 19.5 5:23 1.137 7.23 1:374 6^2 6:8 31.2 26.3 48:2 4.29 23.6 2.81 47.4 7^2 7:9 79:1 1.97 9.13 1:139 1:163 27.7 8^2 8:10 49.2 2.59 35.4 4.41 10.19 9^2 9:11 1:119 47.3 11.15 191:1 10^2 10:12 71.2 2.83 96:2 11^2 11:13 1:167 1.193 12^2 12:14 97.2 13^2 13:15 14^2

I have a truly wonderful proof of the happy factorization theorem, which unfortunately the rest of this page is too small to contain, so I shall have to leave it as an exercise to the reader.

(This is the source for sequences A007966, A007967, A007968, A007969, A007970.)

Received June 28, 1996; published in Journal of Integer Sequences Jan. 1, 1998.

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