{
  "abstract": "For a prime p >= 5 let f_I(p) and f_II(p) count the Type I and Type II solutions of the Erdos-Straus equation 4/p = 1/x + 1/y + 1/z, in the sense of Elsholtz and Tao. We do not address the Erdos-Straus conjecture itself; our subject is the number of solutions. We turn a bijection of Bradford into an exact counting identity: f_I(p) and f_II(p) are sums, over the denominators n coprime to p, of the number of divisors of n^2 in a single residue class modulo e_n = 4n - p. A dual identity puts both counts on the same integer and the same modulus, distinguished only by the classes -p and -1. Our main unconditional result holds for every numerator m: the number of Type II solutions of m/p = 1/x + 1/(pY) + 1/(pZ) with Y <= K is at most sum_{n <= K} tau(n) = K log K + O(K), uniformly in p and m; on average over p <= N it is only O_m((log K)^3 log log K). Hence Type II solutions concentrate at x/p -> 1/m, while Type I solutions cannot. The numerator carries arithmetic information in two independent ways. A real character chi modulo e_n that is trivial on the primes dividing n annihilates Type II when chi(-1) = -1, and Type I when chi(-1)chi(m_0) = -1, where m_0 is the squarefree kernel of m; these coincide exactly when m is a square, so for the numerator 4 no criterion modulo squares can separate the two types. Separately, f_I(n) = f_II(n) = 0 for every odd perfect square n whenever 4 divides m, generalising a theorem of Elsholtz and Tao. Finally we prove the sharp local inequality 2B(n) >= A(n) for e_n in {1,3,5}, with a construction showing it fails for every odd e with 7 <= e <= 2*10^4; we conjecture f_I(p) > f_II(p) for all p > 5 with p not congruent to 1 mod 8, verify this for p <= 10^6, and reduce it to a finite window. We also locate exactly the factor log log N in the Elsholtz-Tao upper bound for the sum of f_I(p).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09204",
  "author_total": 1,
  "authors": [
    {
      "name": "Xinyang Jiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7qe93zsszdf89.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09204",
  "fulltext_url": null,
  "id": "7qe93zsszdf89",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09204",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GM"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7212121553",
  "pdf_url": "https://arxiv.org/pdf/2609.09204",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7qe93zsszdf89.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.09204",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09204"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Analytic Number Theory Research",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Counting Type I and Type II solutions of the Erdos-Straus equation: concentration, local obstructions, and a pointwise comparison",
  "title_text": "Counting Type I and Type II solutions of the Erdos-Straus equation: concentration, local obstructions, and a pointwise comparison",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7qe93zsszdf89",
  "venue": "arXiv",
  "versions": []
}