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Here we obtain the best possible improvement of the bound on $m.$ We prove that if $3|a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+1$ but not when $m=2a,$ and that if $3\\nmid a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+2$ but not when $m=2a+1"},"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":"1306.0775","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-04T13:14:48Z","cross_cats_sorted":[],"title_canon_sha256":"49ffd61836ffa25308feaeb0a7c7f4c9320ebf183801350b1ef3244765cfc9d9","abstract_canon_sha256":"da7870eabf46f5b34db17d0c7c211fae798b52bb73bf08ca1bd6b731979a1f2d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:00:39.438512Z","signature_b64":"0Ok0YbGeXR11b8AVfUtZCOScFfdCVWEKq2pFOreoyeJ4fJYatE78SXOtaxwu4XrJiD4YsWLvoxjoTeAAW5ZzAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6289ee693423db8faaa640065769c92a389e4c11bba15c4f29e995431682e21","last_reissued_at":"2026-05-18T03:00:39.437684Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:00:39.437684Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 2-color Rado Number of $x_1+x_2+\\cdots +x_{m-1}=ax_m,$ II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Saracino","submitted_at":"2013-06-04T13:14:48Z","abstract_excerpt":"In the first installment of this series, we proved that, for every integer $a\\geq 3$ and every $m\\geq 2a^2-a+2$, the 2-color Rado number of $x_1 + x_2 + \\cdots + x_{m-1} = ax_m$ is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$. Here we obtain the best possible improvement of the bound on $m.$ We prove that if $3|a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+1$ but not when $m=2a,$ and that if $3\\nmid a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+2$ but not when $m=2a+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0775","kind":"arxiv","version":3},"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":"1306.0775","created_at":"2026-05-18T03:00:39.437826+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0775v3","created_at":"2026-05-18T03:00:39.437826+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0775","created_at":"2026-05-18T03:00:39.437826+00:00"},{"alias_kind":"pith_short_12","alias_value":"YYUJ5ZUTII63","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"YYUJ5ZUTII63R6VK","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"YYUJ5ZUT","created_at":"2026-05-18T12:28:09.283467+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/YYUJ5ZUTII63R6VKMQAGK5U4SK","json":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK.json","graph_json":"https://pith.science/api/pith-number/YYUJ5ZUTII63R6VKMQAGK5U4SK/graph.json","events_json":"https://pith.science/api/pith-number/YYUJ5ZUTII63R6VKMQAGK5U4SK/events.json","paper":"https://pith.science/paper/YYUJ5ZUT"},"agent_actions":{"view_html":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK","download_json":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK.json","view_paper":"https://pith.science/paper/YYUJ5ZUT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0775&json=true","fetch_graph":"https://pith.science/api/pith-number/YYUJ5ZUTII63R6VKMQAGK5U4SK/graph.json","fetch_events":"https://pith.science/api/pith-number/YYUJ5ZUTII63R6VKMQAGK5U4SK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK/action/storage_attestation","attest_author":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK/action/author_attestation","sign_citation":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK/action/citation_signature","submit_replication":"https://pith.science/pith/YYUJ5ZUTII63R6VKMQAGK5U4SK/action/replication_record"}},"created_at":"2026-05-18T03:00:39.437826+00:00","updated_at":"2026-05-18T03:00:39.437826+00:00"}