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

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 3, page 697-714
  • ISSN: 1246-7405

Abstract

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Let φ denote Euler’s totient function. A century-old conjecture of Carmichael asserts that for every n , the equation φ ( n ) = φ ( m ) has a solution m ≠ n . This suggests defining F ( n ) as the number of solutions m to the equation φ ( n ) = φ ( m ) . (So Carmichael’s conjecture asserts that F ( n ) ≥ 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 ≥ 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 ϵ > 0 , K ( n ) 1 / 2 - ϵ < F ( n ) < K ( n ) 3 / 2 + ϵ for almost all natural numbers n . As an application, we show that φ ( 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.

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 -

References

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