On the Lehmer's problem involving Euler's totient function

TL;DR

This paper proves that no composite number n exists for which Euler's totient function divides n-1, addressing Lehmer's problem, and also discusses the infinitude of primes that are not Sophie Germain primes.

Contribution

The paper resolves Lehmer's problem by showing no composite n satisfies the divisibility condition involving Euler's totient function.

Findings

No composite n with φ(n) dividing n-1 exists

Infinitely many primes are not Sophie Germain primes

Addresses a longstanding open problem in number theory

Abstract

The Euler's totient function $ \varphi(n) $ counts the positive integers up to a given integer $ n$ that are relatively prime to $ n $. We solve a problem due to Lehmer that there is no composite number $ n $ such that $ \varphi(n)\mid n-1 $. We also note that there are infinitely many primes that are not Sophie Germain primes.