{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:4SDHQDTKOIODGT2JN4PBJ4W5M3","short_pith_number":"pith:4SDHQDTK","schema_version":"1.0","canonical_sha256":"e486780e6a721c334f496f1e14f2dd66cc320cab4207efb3a1690ab7d02e205c","source":{"kind":"arxiv","id":"1311.6484","version":1},"attestation_state":"computed","paper":{"title":"The sum of the squares of p positive integers which are consecutive terms of an arithmetic progression: Always a non-perfect square when p(a prime)=3 or p is congruent to 5 or 7 modulo 12","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Konstantine Zelator","submitted_at":"2013-11-21T21:05:21Z","abstract_excerpt":"In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or integer square. In that paper, we made use of the 3-parameter formulas which describe the entire set of positive integer solutions of the 4-variable equation, $x^2+y^2+z^2= t^2$ (See reference [1]) In this work, we offer an alternative proof to the above result; a proof that uses only powers of 3 divisibility arguments. This is done in Theorem1, Section2. After"},"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":"1311.6484","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2013-11-21T21:05:21Z","cross_cats_sorted":[],"title_canon_sha256":"268a6582baa65673c20476ac550b6ced6abb5bf8afab063f590e4527e30d6c95","abstract_canon_sha256":"246534e42dc4ac0c60c4b81ff74cd3cd32631e5ec7536f034d03f8875cb5b442"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:06:12.541919Z","signature_b64":"PbmiHDzkFZv6rjTXkVIEfxQI3fzEaQgRsc17FNwbExVP69CL8H5iZlIuaCaxDbfb/qnISuoZoj5GtRBpnPi+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e486780e6a721c334f496f1e14f2dd66cc320cab4207efb3a1690ab7d02e205c","last_reissued_at":"2026-05-18T03:06:12.541366Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:06:12.541366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The sum of the squares of p positive integers which are consecutive terms of an arithmetic progression: Always a non-perfect square when p(a prime)=3 or p is congruent to 5 or 7 modulo 12","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Konstantine Zelator","submitted_at":"2013-11-21T21:05:21Z","abstract_excerpt":"In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or integer square. In that paper, we made use of the 3-parameter formulas which describe the entire set of positive integer solutions of the 4-variable equation, $x^2+y^2+z^2= t^2$ (See reference [1]) In this work, we offer an alternative proof to the above result; a proof that uses only powers of 3 divisibility arguments. This is done in Theorem1, Section2. After"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.6484","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":"1311.6484","created_at":"2026-05-18T03:06:12.541454+00:00"},{"alias_kind":"arxiv_version","alias_value":"1311.6484v1","created_at":"2026-05-18T03:06:12.541454+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.6484","created_at":"2026-05-18T03:06:12.541454+00:00"},{"alias_kind":"pith_short_12","alias_value":"4SDHQDTKOIOD","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4SDHQDTKOIODGT2J","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4SDHQDTK","created_at":"2026-05-18T12:27:34.582898+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/4SDHQDTKOIODGT2JN4PBJ4W5M3","json":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3.json","graph_json":"https://pith.science/api/pith-number/4SDHQDTKOIODGT2JN4PBJ4W5M3/graph.json","events_json":"https://pith.science/api/pith-number/4SDHQDTKOIODGT2JN4PBJ4W5M3/events.json","paper":"https://pith.science/paper/4SDHQDTK"},"agent_actions":{"view_html":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3","download_json":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3.json","view_paper":"https://pith.science/paper/4SDHQDTK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1311.6484&json=true","fetch_graph":"https://pith.science/api/pith-number/4SDHQDTKOIODGT2JN4PBJ4W5M3/graph.json","fetch_events":"https://pith.science/api/pith-number/4SDHQDTKOIODGT2JN4PBJ4W5M3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3/action/storage_attestation","attest_author":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3/action/author_attestation","sign_citation":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3/action/citation_signature","submit_replication":"https://pith.science/pith/4SDHQDTKOIODGT2JN4PBJ4W5M3/action/replication_record"}},"created_at":"2026-05-18T03:06:12.541454+00:00","updated_at":"2026-05-18T03:06:12.541454+00:00"}