{"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"}