International Journal of Mathematics and Mathematical Sciences
Volume 9 (1986), Issue 4, Pages 801-806
On powerful numbers
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary T2N 1N4, Alberta, Canada
Received 10 December 1985
Copyright © 1986 R. A. Mollin and P. G. Walsh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A powerful number is a positive integer satisfying the property that divides whenever the prime divides ; i.e., in the canonical prime decomposition of , no prime appears with exponent 1. In , S.W. Golomb introduced and studied such numbers. In particular, he asked whether is the only pair of consecutive odd powerful numbers. This question was settled in  by W.A. Sentance who gave necessary and sufficient conditions for the existence of such pairs. The first result of this paper is to provide a generalization of Sentance's result by giving necessary and sufficient conditions for the existence of pairs of powerful numbers spaced evenly apart. This result leads us naturally to consider integers which are representable as a proper difference of two powerful numbers, i.e. where and are powerful numbers with g.c.d. . Golomb (op.cit.) conjectured that is not a proper difference of two powerful numbers, and that there are infinitely many numbers which cannot be represented as a proper difference of two powerful numbers. The antithesis of this conjecture was proved by W.L. McDaniel  who verified that every non-zero integer is in fact a proper difference of two powerful numbers in infinitely many ways. McDaniel's proof is essentially an existence proof. The second result of this paper is a simpler proof of McDaniel's result as well as an effective algorithm (in the proof) for explicitly determining infinitely many such representations. However, in both our proof and McDaniel's proof one of the powerful numbers is almost always a perfect square (namely one is always a perfect square when ). We provide in §2 a proof that all even integers are representable in infinitely many ways as a proper nonsquare difference; i.e., proper difference of two powerful numbers neither of which is a perfect square. This, in conjunction with the odd case in , shows that every integer is representable in infinitely many ways as a proper nonsquare difference. Moreover, in §2 we present some miscellaneous results and conclude with a discussion of some open questions.