PreprintCC BY 4.0arXiv
A Korselt Characterization of Carmichael Numbers of the Form (aCn+1)(aCn+1+1)(aCn+2+1)
Abstract
We study products N=(aCn+1)(aCn+1+1)(aCn+2+1), where (Cn) is the Lucas--balancing sequence. Under the assumption that all three factors are prime, we give a necessary and sufficient condition for N to be a Carmichael number: this happens precisely when n is even and a=kCn+1 for a positive integer k satisfying 48k≡ 7pmodCnCn+2. We derive further congruence restrictions, an explicit parametrization of the integer solutions of the resulting congruence, a lower bound for k, and a divisibility relation involving the middle prime factor. The characterization is conditional on the primality of the three factors and does not assert that such prime triples exist for infinitely many parameters.
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BibTeX
@misc{bensella2026korselt,
title = {{A Korselt Characterization of Carmichael Numbers of the Form $(aC_n+1)(aC_n+1+1)(aC_n+2+1)$}},
author = {Hayat Bensella},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02054},
eprint = {2611.02054},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02054},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Korselt Characterization of Carmichael Numbers of the Form (aC_n+1)(aC_n+1+1)(aC_n+2+1)
AU - Hayat Bensella
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02054
UR - https://arxiv.org/abs/2611.02054
AB - We study products N=(aC_n+1)(aC_n+1+1)(aC_n+2+1), where (C_n) is the Lucas--balancing sequence. Under the assumption that all three factors are prime, we give a necessary and sufficient condition for N to be a Carmichael number: this happens precisely when n is even and a=kC_n+1 for a positive integer k satisfying 48k≡ 7pmodC_nC_n+2. We derive further congruence restrictions, an explicit parametrization of the integer solutions of the resulting congruence, a lower bound for k, and a divisibility relation involving the middle prime factor. The characterization is conditional on the primality of the three factors and does not assert that such prime triples exist for infinitely many parameters.
ER -