{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:G2OVLNDMQSZ2M2DZAT2GBSRHIN","short_pith_number":"pith:G2OVLNDM","schema_version":"1.0","canonical_sha256":"369d55b46c84b3a6687904f460ca274367a8311ae8dd687af345b99647290965","source":{"kind":"arxiv","id":"1609.04391","version":1},"attestation_state":"computed","paper":{"title":"When is $a^{n} + 1$ the sum of two squares?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aaron Schmitt, Emily Stamm, Greg Dresden, Jeremy Rouse, Kylie Hess, Pan Yue, Saimon Islam, Terrin Warren","submitted_at":"2016-09-14T19:50:18Z","abstract_excerpt":"Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares for all $n \\in \\mathbb{N}$ if and only if $a$ is a perfect square. We also prove that for $a\\equiv 0,1,2\\pmod{4},$ if $a^{n} + 1$ is the sum of two squares, then $a^{\\delta} + 1$ is the sum of two squares for all $\\delta | n, \\ \\delta>1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\\equiv 1 \\pmod{4}$, then there are either zero or"},"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":"1609.04391","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-09-14T19:50:18Z","cross_cats_sorted":[],"title_canon_sha256":"8335f259417f1c912b1dc0cfbf28e0114ec07918f8f79cd5030406558890feae","abstract_canon_sha256":"e8de3a8f919b5c9c2f15661e0a7dafbeafb81082f8281c733586e8a5506e2615"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:02.489931Z","signature_b64":"l2XXSqoNEPApPCSxKg2boGbuInHvDchGeEml8ai1iX9KYOgCd9bcvd9oLmKmBoIBdXPrTCTab/Rgcr7PLfI9Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"369d55b46c84b3a6687904f460ca274367a8311ae8dd687af345b99647290965","last_reissued_at":"2026-05-17T23:48:02.489366Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:02.489366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"When is $a^{n} + 1$ the sum of two squares?","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Aaron Schmitt, Emily Stamm, Greg Dresden, Jeremy Rouse, Kylie Hess, Pan Yue, Saimon Islam, Terrin Warren","submitted_at":"2016-09-14T19:50:18Z","abstract_excerpt":"Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n}+1$ can be expressed as the sum of two integer squares. We prove that $a^n + 1$ is the sum of two squares for all $n \\in \\mathbb{N}$ if and only if $a$ is a perfect square. We also prove that for $a\\equiv 0,1,2\\pmod{4},$ if $a^{n} + 1$ is the sum of two squares, then $a^{\\delta} + 1$ is the sum of two squares for all $\\delta | n, \\ \\delta>1$. Using Aurifeuillian factorization, we show that if $a$ is a prime and $a\\equiv 1 \\pmod{4}$, then there are either zero or"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.04391","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":"1609.04391","created_at":"2026-05-17T23:48:02.489431+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.04391v1","created_at":"2026-05-17T23:48:02.489431+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.04391","created_at":"2026-05-17T23:48:02.489431+00:00"},{"alias_kind":"pith_short_12","alias_value":"G2OVLNDMQSZ2","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"G2OVLNDMQSZ2M2DZ","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"G2OVLNDM","created_at":"2026-05-18T12:30:15.759754+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/G2OVLNDMQSZ2M2DZAT2GBSRHIN","json":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN.json","graph_json":"https://pith.science/api/pith-number/G2OVLNDMQSZ2M2DZAT2GBSRHIN/graph.json","events_json":"https://pith.science/api/pith-number/G2OVLNDMQSZ2M2DZAT2GBSRHIN/events.json","paper":"https://pith.science/paper/G2OVLNDM"},"agent_actions":{"view_html":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN","download_json":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN.json","view_paper":"https://pith.science/paper/G2OVLNDM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.04391&json=true","fetch_graph":"https://pith.science/api/pith-number/G2OVLNDMQSZ2M2DZAT2GBSRHIN/graph.json","fetch_events":"https://pith.science/api/pith-number/G2OVLNDMQSZ2M2DZAT2GBSRHIN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN/action/storage_attestation","attest_author":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN/action/author_attestation","sign_citation":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN/action/citation_signature","submit_replication":"https://pith.science/pith/G2OVLNDMQSZ2M2DZAT2GBSRHIN/action/replication_record"}},"created_at":"2026-05-17T23:48:02.489431+00:00","updated_at":"2026-05-17T23:48:02.489431+00:00"}