论文标题
奇怪的，欺骗完美的因素化
Odd, spoof perfect factorizations
论文作者
论文摘要
我们研究了与完美数字相关的二芬太汀方程的整数解决方案。这些解决方案概括了Descartes在1638年发现的示例，其中一个奇数，``欺骗''完美分解$ 3^2 \ cdot 7^2 \ cdot 11^2 \ cdot 13^2 \ cdot 22021^1 $。最近，沃伊特（Voight）发现了欺骗性分解$ 3^4 \ cdot 7^2 \ cdot 11^2 \ cdot 19^2 \ cdot（-127）^1 $。文献中没有其他例子出现。我们计算所有非平凡的，奇怪的，原始的欺骗完美的因素，较少的碱基 - 总共有21个。我们表明，奇怪的，欺骗完美的因素化的结构极为丰富，并且有多个无限家庭。这意味着某些方法解决了奇数的完美数字问题，这些方法仅使用划分函数的乘法性质是不可行的。另一方面，我们证明只有许多非平凡，奇怪的，原始的欺骗性的完美因素，并具有固定数量的碱基。
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.
