The number of solutions of lambda(x)=n

TL;DR

This paper investigates the existence of integers sharing the same Carmichael function value as a given number, providing both unconditional and conditional proofs using prime chain counts and the Extended Riemann Hypothesis.

Contribution

It offers a near-unconditional proof and a complete conditional proof under ERH that for every n, there exists an m with lambda(m)=lambda(n).

Findings

Unconditional near-proof of the existence of such m for each n.

Conditional proof assuming the Extended Riemann Hypothesis.

Utilizes prime chain counting techniques from prior research.

Abstract

We study the question of whether for each n there is another integer m with lambda(m)=lambda(n), where lambda is Carmichael's function. We give a "near" proof of the fact that this is the case unconditionally, and a complete conditional proof under the Extended Riemann Hypothesis. The main tool is a count of prime chains from paper arXiv:0906.3380.