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We show that $s(K) \\ll K \\exp\\left(\\frac{(3\\log 2 + {\\rm o}(1))\\log K}{\\log \\log K}\\right)$ for $K \\geq 2$. This improves on $s(K) \\ll_{\\epsilon} K^{2 +\\epsilon}$, which is the best available upper bound for $s(K)$."},"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":"1710.07767","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-10-21T07:11:51Z","cross_cats_sorted":[],"title_canon_sha256":"e6a704da55a7ec43925569bc4db9a4984d80303e1be773fb888e7d5eb809a864","abstract_canon_sha256":"21f5fb38a463e4248cb2ab713d4ff496d600132a59936772c5bdde1451511ba8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:19.714839Z","signature_b64":"ZAW3jSefgkjEkor2MSx+Ulmlk7aU7jzhpucxFK0axbZh/rNogBWWWY82h77THAw1b5JLVJlhKUaFhAygyUvuBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b173907a1512f06c987bb7988338c46adc293ebc9824c8235745b4592bbe06ae","last_reissued_at":"2026-05-18T00:32:19.714188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:19.714188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On monochromatic representation of sums of squares of primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"D. 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This improves on $s(K) \\ll_{\\epsilon} K^{2 +\\epsilon}$, which is the best available upper bound for $s(K)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.07767","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":""},"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":"1710.07767","created_at":"2026-05-18T00:32:19.714293+00:00"},{"alias_kind":"arxiv_version","alias_value":"1710.07767v1","created_at":"2026-05-18T00:32:19.714293+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1710.07767","created_at":"2026-05-18T00:32:19.714293+00:00"},{"alias_kind":"pith_short_12","alias_value":"WFZZA6QVCLYG","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_16","alias_value":"WFZZA6QVCLYGZGD3","created_at":"2026-05-18T12:31:53.515858+00:00"},{"alias_kind":"pith_short_8","alias_value":"WFZZA6QV","created_at":"2026-05-18T12:31:53.515858+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/WFZZA6QVCLYGZGD3W6MIGOGENL","json":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL.json","graph_json":"https://pith.science/api/pith-number/WFZZA6QVCLYGZGD3W6MIGOGENL/graph.json","events_json":"https://pith.science/api/pith-number/WFZZA6QVCLYGZGD3W6MIGOGENL/events.json","paper":"https://pith.science/paper/WFZZA6QV"},"agent_actions":{"view_html":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL","download_json":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL.json","view_paper":"https://pith.science/paper/WFZZA6QV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1710.07767&json=true","fetch_graph":"https://pith.science/api/pith-number/WFZZA6QVCLYGZGD3W6MIGOGENL/graph.json","fetch_events":"https://pith.science/api/pith-number/WFZZA6QVCLYGZGD3W6MIGOGENL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL/action/storage_attestation","attest_author":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL/action/author_attestation","sign_citation":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL/action/citation_signature","submit_replication":"https://pith.science/pith/WFZZA6QVCLYGZGD3W6MIGOGENL/action/replication_record"}},"created_at":"2026-05-18T00:32:19.714293+00:00","updated_at":"2026-05-18T00:32:19.714293+00:00"}