Odd, spoof perfect factorizations

Abstract

We investigate the integer solutions of Diophantine equations related to perfect numbers. These solutions generalize the example, found by Descartes in 1638, of an odd, “spoof” perfect factorization 32⋅72⋅112⋅132⋅220211. More recently, Voight found the spoof perfect factorization 34⋅72⋅112⋅192⋅(−127)1. No other examples appear in the literature. We compute all nontrivial, odd, primitive spoof perfect factorizations with fewer than seven bases—there are twenty-one in total.We show that the structure of odd, spoof perfect factorizations is extremely rich, and there are multiple infinite families of them. This implies that certain approaches to the odd perfect number problem that use only the multiplicative nature of the sum-of-divisors function are unworkable. On the other hand, we prove that there are only finitely many nontrivial, odd, primitive spoof perfect factorizations with a fixed number of bases; this generalizes previous results, which presupposed positivity of the bases.