The Family Xqps for Distinct Primes q,p in Connection with Carmichael's Totient Conjecture
Abstract
Shannon, Shiue, Saito and the author \cite{SSHS2025} studied the sets Xk={n∈ℕ:φ(n)=k} in connection with Carmichael's totient conjecture, and proved that for an odd prime p≠3 and s∈{1,2,3,4}, X2ps is empty whenever 2ps+1 is not prime (with s=1 fully characterized in both directions), conjecturing that this persists for every s≥1. We resolve that conjecture in full, obtaining a complete characterization of X2ps for every s≥1 and every odd prime p≠3, together with a parity dichotomy: X2ps=∅ unconditionally whenever s is even, while for odd s both outcomes genuinely occur, determined exactly by the primality of 2ps+1. We then close the one remaining gap at p=3, where the proof technique degenerates, with a separate, unconditional characterization of X2·3s. Building on these results, we consider the natural generalization Xqps, where q≠ p are distinct primes and s≥1. We show that this generalization splits into three cases of very different character. If q and p are both odd, the problem is trivial: qps is odd, so Xqps=∅ by elementary parity, with no case-analysis needed. If q=2, the problem is exactly the family X2ps resolved above. If p=2, giving the ``mirror'' family X2sq for an odd prime q, the problem is genuinely different. Beyond s=1, we give a complete resolution for s=2, recovering, with an explicit proof and solution sets, the qualitative dichotomy of \cite[Thm.~2.3.1]{SSHS2025}, but the problem remains open for s≥3.
Related
- Idempotence in Carmichael and Giuga numbers and its application to the Agoh-Giuga conjecture
- An Explicit Family of Log-Concave Counterexamples to the Gaussian Completely Monotone Conjecture
- An infinite family of counterexamples to the Polycirculant Conjecture
- The Guo-Zhang-Qi conjecture and positive Laplace kernels for a digamma power family
- On the noninjectivity of Jordan's totient functions
BibTeX
@misc{he2026family,
title = {{The Family $X_qp^s$ for Distinct Primes $q,p$ in Connection with Carmichael's Totient Conjecture}},
author = {Tian-Xiao He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34808},
eprint = {2610.34808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34808},
}
RIS
TY - UNPB
TI - The Family X_qp^s for Distinct Primes q,p in Connection with Carmichael's Totient Conjecture
AU - Tian-Xiao He
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34808
UR - https://arxiv.org/abs/2610.34808
AB - Shannon, Shiue, Saito and the author \cite{SSHS2025} studied the sets X_k={n∈ℕ:φ(n)=k} in connection with Carmichael's totient conjecture, and proved that for an odd prime p≠3 and s∈{1,2,3,4}, X_2p^s is empty whenever 2p^s+1 is not prime (with s=1 fully characterized in both directions), conjecturing that this persists for every s≥1. We resolve that conjecture in full, obtaining a complete characterization of X_2p^s for every s≥1 and every odd prime p≠3, together with a parity dichotomy: X_2p^s=∅ unconditionally whenever s is even, while for odd s both outcomes genuinely occur, determined exactly by the primality of 2p^s+1. We then close the one remaining gap at p=3, where the proof technique degenerates, with a separate, unconditional characterization of X_2·3^s. Building on these results, we consider the natural generalization X_qp^s, where q≠ p are distinct primes and s≥1. We show that this generalization splits into three cases of very different character. If q and p are both odd, the problem is trivial: qp^s is odd, so X_qp^s=∅ by elementary parity, with no case-analysis needed. If q=2, the problem is exactly the family X_2p^s resolved above. If p=2, giving the ``mirror'' family X_2^sq for an odd prime q, the problem is genuinely different. Beyond s=1, we give a complete resolution for s=2, recovering, with an explicit proof and solution sets, the qualitative dichotomy of \cite[Thm.~2.3.1]{SSHS2025}, but the problem remains open for s≥3.
ER -