On Carmichael's Conjecture

TL;DR

This paper proves that for a fixed n, the equation φ(x)=n has finitely many solutions, characterizes these solutions, and explores the properties of unique solutions, including their prime factorization.

Contribution

It establishes the finiteness of solutions to φ(x)=n for fixed n and provides a general form for these solutions, advancing understanding of Euler's totient function.

Findings

Equation φ(x)=n has finitely many solutions for fixed n

Unique solutions are products of many primes

Conjecture that the number of primes in such solutions is infinite

Abstract

In this article we prove that equation $\phi(x)=n$, for a fixed $n$, admits a finite number of solutions, we find the general form of these solutions, and we show that: if $x_0$ is a unique solution of this equation then $x_0$ is a product of a very large number of primes (we conjecture that the number of such primes is infinite).