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A CONDITIONAL DENSITY FOR CARMICHAEL NUMBERS

Part of: Elementary number theory Multiplicative number theory

Published online by Cambridge University Press:  13 February 2020

THOMAS WRIGHT
THOMAS WRIGHT*
Affiliation:
429 N. Church St., Spartanburg, SC29302, USA email wrighttj@wofford.edu
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Abstract

Under sufficiently strong assumptions about the first prime in an arithmetic progression, we prove that the number of Carmichael numbers up to $X$ is $\gg X^{1-R}$, where $R=(2+o(1))\log \log \log \log X/\text{log}\log \log X$. This is close to Pomerance’s conjectured density of $X^{1-R}$ with $R=(1+o(1))\log \log \log X/\text{log}\log X$.

Keywords

Carmichael numberHeath-Brown conjectureleast prime in an arithmetic progression

MSC classification

Primary: 11A51: Factorization; primality
Secondary: 11N13: Primes in progressions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 101 , Issue 3 , June 2020 , pp. 379 - 388
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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