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In Th.1, we show that eq.(1) if n=8N, N odd; then eq.(1) has no integer solutions; which generalizes problem CC24(the case n=24).\n  We use Th.2, to find some rational solutions of eq.(1); which answers the second question in CC24. In Th.4, we show that","authors_text":"Konstantine Zelator","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2013-08-16T16:51:14Z","title":"Integer Solutions, Rational solutions of the equations x^4+y^4+z^4 -2(x^2)(y^2)-2(y^2)(z^2)-2(z^2)(x^2)=n and x^2+y^4+z^4-2x(y^2)-2x(z^2)-2(y^2)(z^2)=n; And Crux Mathematicorum Contest problem CC24"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4040","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:1828be541003d51c289431516bedf589988c7f6c83720ef48d50c7aa6835530a","target":"record","created_at":"2026-05-18T03:15:35Z","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":"07dfe08246d2b86c395cb48243f4873f0c9e7f10ca3242eb9e0963d6db94c11e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.GM","submitted_at":"2013-08-16T16:51:14Z","title_canon_sha256":"1f10cab1bcd832f5ae8434007f06146e57e8c6b26bfb15dc45dfcf191b456cd6"},"schema_version":"1.0","source":{"id":"1308.4040","kind":"arxiv","version":1}},"canonical_sha256":"fd2f67bfdfd4d5d7b563ba1982ae137d6e459fcf649f9cafdc0d1c09ce866f55","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd2f67bfdfd4d5d7b563ba1982ae137d6e459fcf649f9cafdc0d1c09ce866f55","first_computed_at":"2026-05-18T03:15:35.029255Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:15:35.029255Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/+zPzwEDhph/gBrday8qZaBk0qm91SGToCie6OAUvVLD2DwFhiBFGoMDjz+QpElS2Uhr4x3Qf8C6DGWzK+gmBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:15:35.029858Z","signed_message":"canonical_sha256_bytes"},"source_id":"1308.4040","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1828be541003d51c289431516bedf589988c7f6c83720ef48d50c7aa6835530a","sha256:be959228bc4efbe9a8e46199807da479061cc091c98006aa55918586f512f429"],"state_sha256":"c30dc9280656399ac8655f382e9e558fcaee7ecfcce700fc9e4a2b42cf6a3706"}