Bertrand’s Postulate for Carmichael Numbers

DOI10.1093/IMRN/RNAC203zbMATH Open1522.11095arXiv2111.06963OpenAlexW3213767073MaRDI QIDQ6116636

Publication date: 16 August 2023

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all

d e l t a > 0

and

x

sufficiently large in terms of

d e l t a

, there exist at least

e^{f r a c l o g x (l o g l o g x)^{2 + d e l t a}}

Carmichael numbers between

x

and

x + f r a c x (l o g x)^{f r a c 1 2 + d e l t a}

.

Full work available at URL: https://arxiv.org/abs/2111.06963

zbMATH Keywords

Carmichael numbers Bertrand's postulate

Mathematics Subject Classification ID

Distribution of integers with specified multiplicative constraints (11N25) Factorization; primality (11A51)