An analytical proof for Lehmer's totient conjecture using Mertens' theorems

Abstract

We make an analytical proof for Lehmer's totient conjecture. Lehmer conjectured that there is no solution for the congruence equation $n-1\equiv 0~(mod~\phi(n))$ with composite integers,$n$, where $\phi(n)$ denotes Euler's totient function. He also showed that if the equation has any composite solutions, $n$ must be odd, square-free, and divisible by at least 7 primes. Several people have obtained conditions on values ,$n$, and number of square-free primes constructing $n$ if the equation can have composite solutions. Using Mertens' theorems, we show that it is impossible that the equation can have any composite solution and implies that the conjecture should be true for all the positively composite numbers.