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We conduct a census of $S(\\triangle,\\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \\begin{align*} \\# S(\\triangle,\\square_{2},a)\\cap [x,x+H_{a}\\cdot x^{5/6}\\cdot \\log^{19}x] \\geq x^{5/6-\\varepsilon} \\end{align*} for large $x$. To derive this result and "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2505.23428","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-05-29T13:23:38Z","cross_cats_sorted":[],"title_canon_sha256":"d0dd88a886d0d99e03df88067580a25a8c4b1d02ea4f06964c2ad0dab60016a8","abstract_canon_sha256":"a358bb6d840d94229d225c0b021a82f67e833d543b4ba7b367cac9d22438ff29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:02:05.192557Z","signature_b64":"VrBKCErw05rFSICMTwzPFY2fv2H82FRWRb+2tl7JfoTOsA8c5bj7r1SDEbvAvejp22GqLtZW97DgECYZYHoECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"956f4177122096a9f49e392695c87002488355f7a8859b33cbcab267029b2bb5","last_reissued_at":"2026-05-20T00:02:05.191623Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:02:05.191623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gaps between quadratic forms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Siddharth Iyer","submitted_at":"2025-05-29T13:23:38Z","abstract_excerpt":"Let $\\triangle$ denote the integers represented by the quadratic form $x^2+xy+y^2$ and $\\square_{2}$ denote the numbers represented as a sum of two squares. For a non-zero integer $a$, let $S(\\triangle,\\square_{2},a)$ be the set of integers $n$ such that $n \\in \\triangle$, and $n + a \\in \\square_{2}$. We conduct a census of $S(\\triangle,\\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \\begin{align*} \\# S(\\triangle,\\square_{2},a)\\cap [x,x+H_{a}\\cdot x^{5/6}\\cdot \\log^{19}x] \\geq x^{5/6-\\varepsilon} \\end{align*} for large $x$. To derive this result and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.23428","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/2505.23428/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2505.23428","created_at":"2026-05-20T00:02:05.191776+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.23428v1","created_at":"2026-05-20T00:02:05.191776+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.23428","created_at":"2026-05-20T00:02:05.191776+00:00"},{"alias_kind":"pith_short_12","alias_value":"SVXUC5YSECLK","created_at":"2026-05-20T00:02:05.191776+00:00"},{"alias_kind":"pith_short_16","alias_value":"SVXUC5YSECLKT5E6","created_at":"2026-05-20T00:02:05.191776+00:00"},{"alias_kind":"pith_short_8","alias_value":"SVXUC5YS","created_at":"2026-05-20T00:02:05.191776+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ","json":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ.json","graph_json":"https://pith.science/api/pith-number/SVXUC5YSECLKT5E6HETJLSDQAJ/graph.json","events_json":"https://pith.science/api/pith-number/SVXUC5YSECLKT5E6HETJLSDQAJ/events.json","paper":"https://pith.science/paper/SVXUC5YS"},"agent_actions":{"view_html":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ","download_json":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ.json","view_paper":"https://pith.science/paper/SVXUC5YS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.23428&json=true","fetch_graph":"https://pith.science/api/pith-number/SVXUC5YSECLKT5E6HETJLSDQAJ/graph.json","fetch_events":"https://pith.science/api/pith-number/SVXUC5YSECLKT5E6HETJLSDQAJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ/action/storage_attestation","attest_author":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ/action/author_attestation","sign_citation":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ/action/citation_signature","submit_replication":"https://pith.science/pith/SVXUC5YSECLKT5E6HETJLSDQAJ/action/replication_record"}},"created_at":"2026-05-20T00:02:05.191776+00:00","updated_at":"2026-05-20T00:02:05.191776+00:00"}