# An arithmetic function arising from Carmichael’s conjecture

Florian Luca[1]; Paul Pollack[2]

• [1] Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México
• [2] University of Illinois at Urbana-Champaign Department of Mathematics Urbana, Illinois 61801, USA
• Volume: 23, Issue: 3, page 697-714
• ISSN: 1246-7405

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## Abstract

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Let $\phi$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$, the equation $\phi \left(n\right)=\phi \left(m\right)$ has a solution $m\ne n$. This suggests defining $F\left(n\right)$ as the number of solutions $m$ to the equation $\phi \left(n\right)=\phi \left(m\right)$. (So Carmichael’s conjecture asserts that $F\left(n\right)\ge 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k\ge 2$. Also, the maximal order of $F$ has been investigated by Erdős and Pomerance. In this paper we study the normal behavior of $F$. Let$K\left(x\right):={\left(logx\right)}^{\left(loglogx\right)\left(logloglogx\right)}.$We prove that for every fixed $ϵ>0$,$K{\left(n\right)}^{1/2-ϵ}<F\left(n\right)<K{\left(n\right)}^{3/2+ϵ}$for almost all natural numbers $n$. As an application, we show that $\phi \left(n\right)+1$ is squarefree for almost all $n$. We conclude with some remarks concerning values of $n$ for which $F\left(n\right)$ is close to the conjectured maximum size.

## How to cite

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Luca, Florian, and Pollack, Paul. "An arithmetic function arising from Carmichael’s conjecture." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 697-714. <http://eudml.org/doc/219801>.

@article{Luca2011,
abstract = {Let $\phi$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$, the equation $\phi (n)=\phi (m)$ has a solution $m \ne n$. This suggests defining $F(n)$ as the number of solutions $m$ to the equation $\phi (n)=\phi (m)$. (So Carmichael’s conjecture asserts that $F(n) \ge 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k \ge 2$. Also, the maximal order of $F$ has been investigated by Erdős and Pomerance. In this paper we study the normal behavior of $F$. Let$K(x) := (\log \{x\})^\{(\log \log \{x\})(\log \log \log \{x\})\}.$We prove that for every fixed $\epsilon &gt; 0$,$K(n)^\{1/2-\epsilon \} &lt; F(n) &lt; K(n)^\{3/2+\epsilon \}$for almost all natural numbers $n$. As an application, we show that $\phi (n)+1$ is squarefree for almost all $n$. We conclude with some remarks concerning values of $n$ for which $F(n)$ is close to the conjectured maximum size.},
affiliation = {Instituto de Matemáticas Universidad Nacional Autónoma de México C.P. 58089, Morelia, Michoacán, México; University of Illinois at Urbana-Champaign Department of Mathematics Urbana, Illinois 61801, USA},
author = {Luca, Florian, Pollack, Paul},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Carmichael's conjecture; Euler's function; shifted totients},
language = {eng},
month = {11},
number = {3},
pages = {697-714},
publisher = {Société Arithmétique de Bordeaux},
title = {An arithmetic function arising from Carmichael’s conjecture},
url = {http://eudml.org/doc/219801},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Luca, Florian
AU - Pollack, Paul
TI - An arithmetic function arising from Carmichael’s conjecture
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 697
EP - 714
AB - Let $\phi$ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every $n$, the equation $\phi (n)=\phi (m)$ has a solution $m \ne n$. This suggests defining $F(n)$ as the number of solutions $m$ to the equation $\phi (n)=\phi (m)$. (So Carmichael’s conjecture asserts that $F(n) \ge 2$ always.) Results on $F$ are scattered throughout the literature. For example, Sierpiński conjectured, and Ford proved, that the range of $F$ contains every natural number $k \ge 2$. Also, the maximal order of $F$ has been investigated by Erdős and Pomerance. In this paper we study the normal behavior of $F$. Let$K(x) := (\log {x})^{(\log \log {x})(\log \log \log {x})}.$We prove that for every fixed $\epsilon &gt; 0$,$K(n)^{1/2-\epsilon } &lt; F(n) &lt; K(n)^{3/2+\epsilon }$for almost all natural numbers $n$. As an application, we show that $\phi (n)+1$ is squarefree for almost all $n$. We conclude with some remarks concerning values of $n$ for which $F(n)$ is close to the conjectured maximum size.
LA - eng
KW - Carmichael's conjecture; Euler's function; shifted totients
UR - http://eudml.org/doc/219801
ER -

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