{"paper":{"title":"Multi-target hyperbolic sieves and elliptic trace obstructions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Erik Mulder, Markus Hittmeir, Pantelimon St\\u{a}nic\\u{a}","submitted_at":"2026-06-11T07:53:39Z","abstract_excerpt":"Let $N=pq$ be a semiprime and let $\\ell\\nmid Na$ be an odd prime. The hyperbolic sieve set $H_a(N;\\ell)=\\{ax+Nx^{-1}:x\\in\\mathbb F_\\ell^*\\}$ contains the residue of the linear form $ap+q$ modulo $\\ell$ and has exact cardinality $(\\ell+\\chi(aN))/2$, where $\\chi$ is the Legendre symbol modulo $\\ell$. We study simultaneous sieving for several linear forms and give a complete local analysis of the two-target primitive-root case proposed in connection with deterministic integer factorization. For two distinct coefficients $a,b$, with $A=4aN$ and $B=4bN$, we prove an exact formula for $|H_a(N;\\ell)\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.13018","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.13018/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}