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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","authors_text":"Artyom Radomskii","cross_cats":[],"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.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","title":"On sums of two squares and a basis of order $2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.20653","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-09T23:05:04.301176Z","id":"40d3e1fd-4116-4b13-9f38-80c50c14211e","model_set":{"reader":"grok-4.3"},"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","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.","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.","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."}},"verdict_id":"40d3e1fd-4116-4b13-9f38-80c50c14211e"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282","target":"record","created_at":"2026-05-26T02:04:11Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"769109c5a53bd119e4ed9dc12e7aee3ee38aa006a52edbb742891e12efc70ca6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-04-22T15:03:43Z","title_canon_sha256":"8ea1ed3ee870ab73d575a2707fb5f52d8bead08c41e884785017d92935fa5857"},"schema_version":"1.0","source":{"id":"2604.20653","kind":"arxiv","version":4}},"canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eae70f04b5ad24cd49d887097b0506b5ede00c421a01a8a65c1e3f8de077c913","first_computed_at":"2026-05-26T02:04:11.176201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-26T02:04:11.176201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8IW3rGOsQPoOURH2+mHWIyZR6SERTQ0NbG376caLJ7KUfS1A9x6JHgrXJxcRGom5gSKNDNKXY74SBw/oCXWADg==","signature_status":"signed_v1","signed_at":"2026-05-26T02:04:11.176968Z","signed_message":"canonical_sha256_bytes"},"source_id":"2604.20653","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:22969c164c9576bec12c840a18a76652b3687092b5b0bcd923c3ad383191e282","sha256:6737dc30b9ad79b5f7e57fc56c4b7c80d02145bb64b67eedeef71a0fee0b56d6"],"state_sha256":"056cf98d42711788838cffa185796e544b50f95294d3147bc8305dfd80cb2b9a"}