{
  "abstract": "Let $N=pq$ be a product of two primes, and let $\\ell$ be an odd prime not dividing $N$. For $a\\in\\mathbb F_\\ell^*$, the set \\[ H_a(N;\\ell)=\\{ax+Nx^{-1}:x\\in\\mathbb F_\\ell^*\\} \\] contains the residue of $ap+q$ modulo $\\ell$. Since residues outside this set can be excluded when searching for that linear form, it has been applied as a sieve for linear combinations of $p$ and $q$ in recent deterministic integer factorization methods. Its size is $(\\ell+χ(aN))/2$, where $χ$ is the Legendre symbol modulo $\\ell$. We study unions of these sets, which give a common set of candidate residues for several linear forms and thus allow to sieve for multiple target forms at once. For distinct $a,b$, we obtain an exact formula for $|H_a(N;\\ell)\\cup H_b(N;\\ell)|$ in terms of \\[ K(A,B)=\\sum_{z\\in\\mathbb F_\\ell}χ((z^2-A)(z^2-B)), \\qquad A=4aN,\\quad B=4bN. \\] The Hasse-Weil bound gives an error of $O(\\sqrt\\ell)$ from $3\\ell/4$. For fixed $c\\ne0,1$, we prove the elementary identity $K(A,cA)=-1-χ(A)t_c$, where $t_c=-1-K(1,c)$. Hence the union for a primitive-root pair $(a,b)=(γ,γ^2)$ has at most two sizes as $N$ varies. We give an exact condition on $t_γ$ for the bound $3\\ell/4+1$ to hold for every nonzero $N$ modulo $\\ell$, and compute the first and second moments. For $k$ distinct coefficients, we prove \\[ \\left|\\left|\\bigcup_{j=1}^k H_{a_j}(N;\\ell)\\right| -\\ell(1-2^{-k})\\right| \\le (k-1+2^{-k})\\sqrt\\ell+k. \\] We derive the corresponding bounds modulo products of distinct primes using the Chinese remainder theorem. We also determine the exact size of the image obtained by restricting $x$ to the nonzero $r$-th powers, which applies when searching for a divisor of the form $u^r$ in $N=u^rv$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.13018",
  "author_total": 3,
  "authors": [
    {
      "name": "Pantelimon Stănică"
    },
    {
      "name": "Erik Mulder"
    },
    {
      "name": "Markus Hittmeir"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bdyg6dff0zjhz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.13018",
  "fulltext_url": null,
  "id": "bdyg6dff0zjhz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.13018",
  "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": "W7164560480",
  "pdf_url": "https://arxiv.org/pdf/2606.13018",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bdyg6dff0zjhz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2606.13018",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.13018"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Multi-target hyperbolic sieves and elliptic trace obstructions",
  "title_text": "Multi-target hyperbolic sieves and elliptic trace obstructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/bdyg6dff0zjhz",
  "venue": "arXiv",
  "versions": []
}