On powerful numbers

Abstract

A powerful number is a positive integer n satisfying the property that p2 divides n whenever the prime p divides n; i.e., in the canonical prime decomposition of n, no prime appears with exponent 1. In [1], S.W. Golomb introduced and studied such numbers. In particular, he asked whether (25, 27) is the only pair of consecutive odd powerful numbers. This question was settled in [2] 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. n = p1 − p2 where p1 and p2 are powerful numbers with g.c.d. (p1, p2) = 1. Golomb (op.cit.) conjectured that 6 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 [3] 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 n≢2(mod4)). 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 [4], 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.