數學通報1953年11月號問题與解答欄第68題:「試找出一組正整數a,b,c,滿足方程a~3+b~4=c~5」是有解的,例如:a=31~5,b=31~4c=31~3·2便是一解,不僅如此,我們還可進而證明下列一個較帶普遍性的結論: p,q,r爲正整數,且pr舆q互質,則方程a~p+b~q=c~r有正整數解。下面,我們就叙述這個結論的證明。 (1)先證明一個预備定理:m,n爲互質的正整數,則必有正整數x,y存在,滿足等式: xm-yn=1 證:由代數學知,對於任二個互質的整數 The Mathematics Bulletin of November 1953, Issue No. 68, Question and Answer column 68: “Try to find a set of positive integers a, b, c, satisfy the equation a~3+b~4=c~5”. There are solutions, for example: A=31~5, b=31~4c=31~3·2 is a solution. Not only that, we can prove the following more general conclusion: p, q, r are positive integers, and pr If 舆q is relatively prime, then the equation a~p+b~q=c~r has a positive integer solution. Below, we describe the proof of this conclusion. (1) First prove a preparation theorem: m, n is a coprime positive integer, then there must be a positive integer x, y exists to satisfy the equation: xm-yn=1 proof: known by algebra, for any two Coprime integer