This one is pretty good. Kaplansky wanted squarefree numbers $x$ such that $$ \sigma ( x^3) = y^2, $$ where $\sigma$ is the sum of divisors function. Somewhere around here I have a short note of his. He referred to this (and some very similar problems) as Ozanam's problem, from page 56 of Dickson's History, volume 1. Let's see; for each prime $p$ up to some bound, I had the computer factor $ \sigma ( p^3),$ especially recording the exponents on the output primes $q.$ So, to the best of my memory (about 18 years ago), I wound up with a big matrix with entries in the field with two elements; each column meant a prime $p,$ each row was saying whether the exponent for the prime $q$ was even or odd. Then, a solution was a column vector, also of 0's and 1's, which my big matrix mapped to the zero vector. So, I did Gauss elimination over the field of two elements. And found hundreds of solutions.
I will see if I can find something written about this. For that matter, the program or programs I wrote should still be there in my MSRI account.
Note: i am having a little trouble remembering if it is the matrix i describe above or its transpose. So perhaps a little care is needed. It definitely worked, though, and quickly. I also built in some procedure where i could force some prime to be included, then see if i could find solutions with that restriction. As I recall, that needed more handholding for the computer, more attention by me.
