{
  "abstract": "Shannon, Shiue, Saito and the author \\cite{SSHS2025} studied the sets $X_k=\\{n\\in\\mathbb{N}:φ(n)=k\\}$ in connection with Carmichael's totient conjecture, and proved that for an odd prime $p\\neq3$ and $s\\in\\{1,2,3,4\\}$, $X_{2p^s}$ is empty whenever $2p^s+1$ is not prime (with $s=1$ fully characterized in both directions), conjecturing that this persists for every $s\\ge1$. We resolve that conjecture in full, obtaining a complete characterization of $X_{2p^s}$ for every $s\\ge1$ and every odd prime $p\\neq3$, together with a parity dichotomy: $X_{2p^s}=\\varnothing$ 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 $X_{2\\cdot3^s}$. Building on these results, we consider the natural generalization $X_{qp^s}$, where $q\\neq p$ are distinct primes and $s\\ge1$. 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 $X_{qp^s}=\\varnothing$ 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 $s\\ge3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34808",
  "author_total": 1,
  "authors": [
    {
      "name": "Tian-Xiao He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/157178gxk0ft8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34808",
  "fulltext_url": null,
  "id": "157178gxk0ft8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34808",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7223932718",
  "pdf_url": "https://arxiv.org/pdf/2610.34808",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/157178gxk0ft8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34808",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34808"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The Family $X_{qp^s}$ for Distinct Primes $q,p$ in Connection with Carmichael's Totient Conjecture",
  "title_text": "The Family X_qp^s for Distinct Primes q,p in Connection with Carmichael's Totient Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/157178gxk0ft8",
  "venue": "arXiv",
  "versions": []
}