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; this generalizes previous results, which presupposed positivity of the bases.

我们研究了与完美数相关的丢番图方程的整数解。这些解决方案概括了笛卡尔在 1638 年发现的奇怪的“欺骗”完美分解的例子$3^2\cdot 7^2\cdot {11}^2\cdot {13}^2\cdot {22021}^1$. 最近，Voight 发现了完美分解$3^4\cdot 7^2\cdot {11}^2\cdot {19}^2\cdot {(-127)}^1$. 文献中没有其他例子。我们用少于 7 个碱基计算所有非平凡的、奇数的、原始的欺骗完美分解——总共有 21 个。

我们证明了奇数、恶搞完美因式分解的结构非常丰富，并且它们有多个无限的族。这意味着某些仅使用除数之和函数的乘法性质的奇完美数问题的方法是行不通的。另一方面，我们证明只有有限多个具有固定基数的非平凡、奇数、原始欺骗完美分解；这概括了以前的结果，这些结果以碱基的阳性为前提。