短埃及分数

短埃及分数

Short Egyptian fractions

OpenAI 数学仓库收录的研究手稿。展开查看英文摘要，或标记后让我进一步解释。

具体内容

稿件研究有理数表示为互异单位分数之和的项数问题。作者声称证明了Erdős的一个猜想：对充分大的整数b，每个满足1≤a<b的有理数a/b都可写成O(log log b)个互异正单位分数之和，隐含常数绝对；当分子变化时该阶最优。作者还称，恰含k个互异项的1的展开个数，以及至少为2且从不出现在此类展开分母中的最小整数，均随k双指数增长，其双对数阶为k。

• 作者声称证明Erdős猜想：a/b可写成O(log log b)个互异单位分数之和
• 该阶在分子变化时最优
• 1的k项展开个数与最小缺席分母均随k双指数增长

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英文原始摘要

We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number $`a/b`$ with $`1\le a\lt b`$ is a sum of $`O(\log\log b)`$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.

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