Weird numbers are abundant numbers that are not pseudoperfect. Since their introduction, the existence of odd weird numbers has been an open problem. In this work, we describe our computational effort to search for odd weird numbers, which shows their non-existence up to $10^{21}$. We also searched up to $10^{28}$ for numbers with an abundance below $10^{14}$, to no avail. Our approach to speed up the search can be viewed as an application of reverse search in the domain of combinatorial optimization, and may be useful for other similar quest for natural numbers with special properties that depend crucially on their factorization.
翻译:奇怪的数字是数量众多的, 并不是伪造的。 自开始以来, 奇数的存在就是一个尚未解决的问题。 在这项工作中, 我们描述了我们寻找奇数的计算努力, 这表明它们不存在高达 10 ⁇ 21 美元。 我们还搜索了高达 10 ⁇ 28 美元的数字, 其丰度低于 10 ⁇ 14 美元, 却无济于事。 我们加快搜索的方法可以被视为在组合优化领域进行反向搜索的一种应用, 并且可能有益于其他类似的自然数字搜索, 而这些自然数字的特殊特性主要取决于它们的因子化。