本文利用如下的一个简单等式m个m~n m~n+m~n+…m~n=m~(n+1),求一类不定方程的一个正整数解。例1 求方程x~2+y~3=z~3的一个正整数解,并证明此方程有无穷多个正整数解。解:因为2和3的最小公倍数是6,将原方程与2~n+2~n=2~(n+1)比较,易知既是6的倍数,又比5的倍数小1的最小正整数n的值为24。∵ 2~(24)+2~(24)=2~(25),即 (2~(12))~2+(2~8)~3=(2~5)~5, ∴(2~(12),2~8,2~5)是原方程的一个正整数 In this paper, we use the following simple equation m m~n m~n+m~n+...m~n=m~(n+1) to find a positive integer solution for a class of indefinite equations. Example 1 Find a positive integer solution of the equation x~2+y~3=z~3 and prove that this equation has infinitely many positive integer solutions. Solution: Because the least common multiple of 2 and 3 is 6, comparing the original equation with 2~n+2~n=2~(n+1), it is easy to know that it is a multiple of 6, and is less than the multiple of 5 The value of the integer n is 24. ∵ 2~(24)+2~(24)=2~(25), ie (2~(12))~2+(2~8)~3=(2~5)~5, ∴(2~ (12), 2~8, 2~5) is a positive integer of the original equation