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

TL;DR

This paper provides an analytical proof, using Mertens' theorems, that Lehmer's totient conjecture has no composite solutions, supporting the conjecture's validity for all positive composite numbers.

Contribution

The paper offers a novel proof leveraging Mertens' theorems to confirm Lehmer's conjecture for all composite integers.

Findings

No composite solutions to Lehmer's equation exist.

Lehmer's conjecture holds for all positive composite numbers.

The proof rules out the existence of composite solutions using number theory 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.