On Euler’s Equationφ(x)=k

On Euler's Equation (x)= k

本文研究了Ｅｕｌｅｒ方程（ｘ）＝ｋ的解，我们用Ｓｅｌｂｅｒｇ筛法证明了下述定理：设ｍ，ｋ是任意的正整数，则使方程ｍｐｋ＝（ｙ）有解的不超过ｘ的素数ｐ的个数为Ｏ（ｘ／ｌｏｇ２ｘ）．

A positive integer k is called nontotient if Euler's equation (x)= k has no solution. In this paper, by using Selberg's upper bound sieve method, we obtain a new result: Let m, k be arbitary positive integers, then the number of primes px such that mp k=(y) has solution is O(x/ log 2 x) .