  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 $3^2\cdot 7^2\cdot 11^2\cdot
13^2\cdot 22021^1$. More recently, Voight found the spoof perfect factorization
$3^4\cdot 7^2\cdot 11^2\cdot 19^2\cdot(-127)^1$. 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.
