PreprintCC BY 4.0arXiv
An Asymptotic for Non-Wieferich Numbers
Abstract
A cyclic number is a positive integer n such that gcd(n, φ(n)) = 1. Erdős showed that the number of cyclic numbers up to x, C(x), has C(x) ∼ e−γx/log3(x) asymptotically as x tends to infinity. We define Wieferich numbers to be odd, positive integers n such that (n, (2ℓ(n)−1)/n) > 1, where ℓ(n) is the smallest positive integer with 2ℓ(n) ≡ 1 mod n. Franco and Pomerance showed that non-Wieferich numbers have density zero among the odd integers. We show that the counting function of n ≤ x in the symmetric difference of non-Wieferich numbers and cyclic numbers is O(x/(log3(x))2), under a mild assumption on the counting function of Wieferich primes.
Related
BibTeX
@misc{siegel2026asymptotic,
title = {{An Asymptotic for Non-Wieferich Numbers}},
author = {Casia Siegel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28685},
eprint = {2610.28685},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28685},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Asymptotic for Non-Wieferich Numbers
AU - Casia Siegel
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.28685
UR - https://arxiv.org/abs/2610.28685
AB - A cyclic number is a positive integer n such that gcd(n, φ(n)) = 1. Erdős showed that the number of cyclic numbers up to x, C(x), has C(x) ∼ e^−γx/log_3(x) asymptotically as x tends to infinity. We define Wieferich numbers to be odd, positive integers n such that (n, (2^ℓ(n)−1)/n) > 1, where ℓ(n) is the smallest positive integer with 2^ℓ(n) ≡ 1 mod n. Franco and Pomerance showed that non-Wieferich numbers have density zero among the odd integers. We show that the counting function of n ≤ x in the symmetric difference of non-Wieferich numbers and cyclic numbers is O(x/(log_3(x))^2), under a mild assumption on the counting function of Wieferich primes.
ER -