{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:5LTQ6BFVVUSM2SOYQ4EXWBIGWX","short_pith_number":"pith:5LTQ6BFV","schema_version":"1.0","canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","source":{"kind":"arxiv","id":"2604.20653","version":4},"attestation_state":"computed","paper":{"title":"On sums of two squares and a basis of order $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artyom Radomskii","submitted_at":"2026-04-22T15:03:43Z","abstract_excerpt":"Let $\\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \\nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\\{n_1-m,\\ldots, n_1+m\\}$ and $I_{2}=\\{n_2-m, \\ldots, n_2+m\\}$ such that $m = [(\\log N) (\\log \\log N)^{1/325565}]$, $I_{1}\\cup I_{2} \\subset [1, N]$, $N = n_1 + n_2$, and for any $n\\in I_{1}\\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\\mathcal{R}$. In particular, we have $n(an+b)\\notin \\mathcal{R"},"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":"2604.20653","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","cross_cats_sorted":[],"title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857","abstract_canon_sha256":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:11.176968Z","signature_b64":"8IW3rGOsQPoOURH2+mHWIyZR6SERTQ0NbG376caLJ7KUfS1A9x6JHgrXJxcRGom5gSKNDNKXY74SBw/oCXWADg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","last_reissued_at":"2026-05-26T02:04:11.176201Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:11.176201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sums of two squares and a basis of order $2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Artyom Radomskii","submitted_at":"2026-04-22T15:03:43Z","abstract_excerpt":"Let $\\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \\nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\\{n_1-m,\\ldots, n_1+m\\}$ and $I_{2}=\\{n_2-m, \\ldots, n_2+m\\}$ such that $m = [(\\log N) (\\log \\log N)^{1/325565}]$, $I_{1}\\cup I_{2} \\subset [1, N]$, $N = n_1 + n_2$, and for any $n\\in I_{1}\\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\\mathcal{R}$. In particular, we have $n(an+b)\\notin \\mathcal{R"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7a73aeaa8cbad0a6faac9c59c89fdadcdd68562df35d4e6802a86b9fb2077b78"},"source":{"id":"2604.20653","kind":"arxiv","version":4},"verdict":{"id":"40d3e1fd-4116-4b13-9f38-80c50c14211e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-09T23:05:04.301176Z","strongest_claim":"For sufficiently large positive integer N there are two strings of consecutive positive integers I1={n1-m,…,n1+m} and I2={n2-m,…,n2+m} such that m=[(log N)(log log N)^{1/325565}], I1∪I2⊂[1,N], N=n1+n2, and for any n∈I1∪I2 at least one of n or an+b does not lie in R.","one_line_summary":"For large N there exist paired intervals I1 and I2 of length ~log N (log log N)^{1/325565} with n1 + n2 = N where no n has both n and an+b as primitive sums of two squares.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The existence holds only for sufficiently large N; the specific tiny exponent 1/325565 is an effective constant arising from analytic estimates whose validity for all large N is assumed but not verified in the abstract.","pith_extraction_headline":"For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.20653/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T14:35:22.657108Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-20T01:41:15.484825Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"426387748644c9669d54b9eb99f3529e9543cdece8000d74882613706545b67f"},"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":"2604.20653","created_at":"2026-05-26T02:04:11.176271+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.20653v4","created_at":"2026-05-26T02:04:11.176271+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.20653","created_at":"2026-05-26T02:04:11.176271+00:00"},{"alias_kind":"pith_short_12","alias_value":"5LTQ6BFVVUSM","created_at":"2026-05-26T02:04:11.176271+00:00"},{"alias_kind":"pith_short_16","alias_value":"5LTQ6BFVVUSM2SOY","created_at":"2026-05-26T02:04:11.176271+00:00"},{"alias_kind":"pith_short_8","alias_value":"5LTQ6BFV","created_at":"2026-05-26T02:04:11.176271+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/5LTQ6BFVVUSM2SOYQ4EXWBIGWX","json":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX.json","graph_json":"https://pith.science/api/pith-number/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/graph.json","events_json":"https://pith.science/api/pith-number/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/events.json","paper":"https://pith.science/paper/5LTQ6BFV"},"agent_actions":{"view_html":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX","download_json":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX.json","view_paper":"https://pith.science/paper/5LTQ6BFV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.20653&json=true","fetch_graph":"https://pith.science/api/pith-number/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/graph.json","fetch_events":"https://pith.science/api/pith-number/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/action/storage_attestation","attest_author":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/action/author_attestation","sign_citation":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/action/citation_signature","submit_replication":"https://pith.science/pith/5LTQ6BFVVUSM2SOYQ4EXWBIGWX/action/replication_record"}},"created_at":"2026-05-26T02:04:11.176271+00:00","updated_at":"2026-05-26T02:04:11.176271+00:00"}