一个古埃及问题：丢番图方程4/n=1/x+1/y+1/z，n>或=2
An ancient Egyptian problem:the diophantine equation 4/n=1/x+1/y+1/z, n>or=2
摘要：
从莱因德纸草和其他现存的资料中，我们知道古埃及人非常热衷于将给定的分数表示为单位分数的总和，即分子等于1的分数。在过去的60年里，我们遇到了一个问题，被称为鄂尔多斯-施特劳斯猜想，它指出，对于每个正整数n>1，分数4/n可以分解为三个不同的单位分数之和。自1950年以来，已取得了一些部分结果，见参考文献[1]- [8]，以及[10]和[11]。在这项工作中，我们贡献四个定理。在定理2中，我们证明了如果分数4/n不等于三个不同的单位分数之和，则n的每个素除子必全等于1模24。此外，如果n包含一个与1 mod 24不全等的除数，那么这样的分解确实存在，并且是显式给出的。
英文摘要：
From the Rhind Papyrus and other extant sources, we know that the ancient Egyptians were very iterested in expressing a given fraction into a sum of unit fractions, that is fractions whose numerators are equal to 1. One of the problems that has come down to us in the last 60 years, is known as the Erdos- Strauss conjecture which states that for each positive integer n>1; the fraction 4/n can be decomposed into a sum of three distinct unit fractions. Since 1950, a numberof partial results have been achieved, see references [1]- [8]; and also [10] and[11]. In this work we contribute four theorems. In Theorem 2, we prove that if the fraction 4/n is not equal to a sum of three distinct unit fractions, then each prime divisor of n; must be congruent to 1 modulo24. Moreover, if n contains a divisor noncongruent to 1mod24; then such a decomposition does exist and it is explicitly given.
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