{
  "abstract": "We give an elementary proof that every positive rational number $a/b$ with $b$ squarefree is a finite sum of distinct unit fractions $1/n$, where each $n$ is a product of two distinct primes (Erdős Problem #306). After a reduction to small targets, we take a single complete bipartite graph between the primes in $(y^2,2y^2]$, together with $2$ and the primes of $b$, and a tuned initial segment of the primes in $(y^8,y^9]$, and show that some subgraph has reciprocal sum congruent to $a/b$ modulo $1$; the small total mass then forces equality. Writing the number of such subgraphs as a finite Fourier sum, we sort the frequencies into three cases using a table indexed by the two sides of the graph. The small integer frequencies give a positive main term, and all other frequencies are negligible by a divisor-counting argument and a no-wrap-around form of the Chinese remainder theorem. The only inputs about primes are Chebyshev-type bounds. The circle-method framework comes from Tang's Lean development, which gave the first proof; our construction removes its anchor-synchronisation step. The proof has been formalised in Lean 4, apart from a cited inequality of Ramanujan. This work is a human-AI collaboration: AI tools contributed substantially to the construction, the experiments and the writing.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32140",
  "author_total": 1,
  "authors": [
    {
      "name": "Shisheng Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ahj9y8sbmfs4d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32140",
  "fulltext_url": null,
  "id": "ahj9y8sbmfs4d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32140",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214728778",
  "pdf_url": "https://arxiv.org/pdf/2609.32140",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ahj9y8sbmfs4d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32140",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32140"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306",
  "title_text": "Unit fractions with semiprime denominators: an elementary proof of Erdős Problem #306",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ahj9y8sbmfs4d",
  "venue": "arXiv",
  "versions": []
}