第六个费马素数存在性的一个等价命题

A new necessary and sufficient condition for the existence of the sixth Fermat prime

在数论中,是否存在第六个费马素数一直是个未解决的问题.通过运用欧拉函数建立了第六个费马素数存在性的一个等价命题.也就是,第六个费马素数存在,当且仅当存在一个正整数使得方程k(x)2有34个解,其中(x)为欧拉函数.

The question whether there exists the sixth Fermat prime is still an open problem in number theory. This paper obtains a new necessary and sufficient condition for the existence of the sixth Fermat prime based on Euler＇s totient function. It proves that the sixth Fermat prime exists if and only if there is a positive integer k such that the number of solutions of the equation φ（x） = 2^k is 34.