关于本原商高数的Miyazaki猜想

设a,b,c是适合a=2~(2r)-n~2,b=2~(r+1)n,c=2~(2r)+n~2的正整数,其中r是正整数,n是奇素数.运用初等数论方法讨论了指数Diophantine方程c~x+b~y=a~z.证明了:当2~r=n+1时,方程仅有正整数解(x,y,z)=(1,1,2);否则,方程无解。上述结果部分地证实了有关本原商高数的Miyazaki猜想。

The Miyazaki Conjecture on Primitive Pythagorean Numbers

Let a,b,c be positive integers such that a = 2~(2r) — n~2,b = 2~(r+1)n,c = 2~(2r) + n~(2)where r is a positive integer,n is an odd prime.In this paper,using some elementary number theory methods,the exponential diophantine equation c~x+ b~y = a~z is discussed.We prove that if 2~r = n + 1,then the equation has only the positive integer solution(x,y,z) =(1,1,2),otherwise,it has no solution.This result partly verifies Miyazaki's conjecture on primitive Pythagorean numbers.