{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:OWOIBHOUOQMIOFETMDDPPPNFIM","short_pith_number":"pith:OWOIBHOU","canonical_record":{"source":{"id":"2605.13460","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T12:52:46Z","cross_cats_sorted":[],"title_canon_sha256":"a21eacb5302313c91af3f3b33b8bde935783f262fb92b84acc245a27e37e5c1a","abstract_canon_sha256":"b4eddb356a8a5d555717ca09da4945e4945e94e8c22e621dc8542bc34701ecf8"},"schema_version":"1.0"},"canonical_sha256":"759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4","source":{"kind":"arxiv","id":"2605.13460","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13460","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13460v1","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13460","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"pith_short_12","alias_value":"OWOIBHOUOQMI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"OWOIBHOUOQMIOFET","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"OWOIBHOU","created_at":"2026-05-18T12:33:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:OWOIBHOUOQMIOFETMDDPPPNFIM","target":"record","payload":{"canonical_record":{"source":{"id":"2605.13460","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T12:52:46Z","cross_cats_sorted":[],"title_canon_sha256":"a21eacb5302313c91af3f3b33b8bde935783f262fb92b84acc245a27e37e5c1a","abstract_canon_sha256":"b4eddb356a8a5d555717ca09da4945e4945e94e8c22e621dc8542bc34701ecf8"},"schema_version":"1.0"},"canonical_sha256":"759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:41.719081Z","signature_b64":"qEVWQMMrKoiSVqXFiTsbEVwXs5SiMGU3GyitljcHUg890fhUHmAtY968lQlIaeDU/QeHiBIICeYSp0hBPmKmBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4","last_reissued_at":"2026-05-18T02:44:41.718615Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:41.718615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2605.13460","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1ZzC4S8wY/2BdJ3qTWYLpmGngtj/9b3JbfAdRPJ3bIxX/m9dksXBE8q8fbWhgJ6hDYGilyWEpn+QCg7NHVsmDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:34:52.122622Z"},"content_sha256":"330531c1cf9dd9e7b8a4a2c230c4ebc2d68c0b70678d3f3e4d870140bb2791e4","schema_version":"1.0","event_id":"sha256:330531c1cf9dd9e7b8a4a2c230c4ebc2d68c0b70678d3f3e4d870140bb2791e4"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:OWOIBHOUOQMIOFETMDDPPPNFIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wieferich Primes and Monogenic Trinomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Lenny Jones","submitted_at":"2026-05-13T12:52:46Z","abstract_excerpt":"A prime $p$ is called a Wieferich prime if $2^{p-1}\\equiv 1 \\pmod{p^2}$.\n  A monic polynomial $f(x)\\in {\\mathbb Z}[x]$ of degree $N\\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\\mathbb Q}$ and\n  $\\{1,\\theta,\\theta^2,\\ldots,\\theta^{N-1}\\}$\n  is a basis for the ring of integers of ${\\mathbb Q}(\\theta)$, where $f(\\theta)=0$. In this article, we show that ${\\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"we show that F_p(x):=x^{2p}+2x^{p}+2 is monogenic if and only if p is not a Wieferich prime.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The standard definitions of monogenic polynomials and Wieferich primes are assumed to apply directly, along with the irreducibility of F_p(x) over Q for prime p (details of the proof not visible in abstract).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c7af62e100f6350fb28103698cee064b51eb445d9999705769c82a839a76ea4e"},"source":{"id":"2605.13460","kind":"arxiv","version":1},"verdict":{"id":"611ec171-100e-47b9-8f6b-c9c6730fe4be","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:19:53.597455Z","strongest_claim":"we show that F_p(x):=x^{2p}+2x^{p}+2 is monogenic if and only if p is not a Wieferich prime.","one_line_summary":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The standard definitions of monogenic polynomials and Wieferich primes are assumed to apply directly, along with the irreducibility of F_p(x) over Q for prime p (details of the proof not visible in abstract).","pith_extraction_headline":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich."},"references":{"count":72,"sample":[{"doi":"","year":2000,"title":"Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag , 2000","work_id":"18906986-5d90-4d2f-a55e-c24246f72c28","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"J. Harrington and L. Jones, A note on generalised Wall-Sun-Sun primes , Bull. Aust. Math. Soc. 108 , No. 3, 373--378 (2023)","work_id":"4884c648-debc-4f3a-ba36-c0a9728a08ca","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"A. Jakhar, S. Khanduja and N. Sangwan, Characterization of primes dividing the index of a trinomial, Int. J. Number Theory 13 (2017), no. 10, 2505--2514","work_id":"ca49dad4-133f-4833-8f9d-82da90952231","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"Jones, Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials , Albanian J","work_id":"4ce457e6-76f4-43bc-be0a-4743028f2e85","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Jones, A new condition for k -Wall-Sun-Sun primes , Taiwanese J","work_id":"2359ac34-1500-4526-8b94-26d0434b552b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":72,"snapshot_sha256":"33af78b26a99569bf50db5317d09e8a881051f307fc75ae57311abc6f4027d5f","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"},"verdict_id":"611ec171-100e-47b9-8f6b-c9c6730fe4be"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yh0iXsUG3RjiBIJvGXS466JlW/xdIg7I34WW5rIC2UgK7NUAURjfZlaq9LMaIv1LYXfMQhexZjwm1bEak0fNDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T15:34:52.123437Z"},"content_sha256":"450818438555ed1c7048c82643b638279c225dcfe6a302f25e992b889dc21117","schema_version":"1.0","event_id":"sha256:450818438555ed1c7048c82643b638279c225dcfe6a302f25e992b889dc21117"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/bundle.json","state_url":"https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-31T15:34:52Z","links":{"resolver":"https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM","bundle":"https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/bundle.json","state":"https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OWOIBHOUOQMIOFETMDDPPPNFIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:OWOIBHOUOQMIOFETMDDPPPNFIM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"b4eddb356a8a5d555717ca09da4945e4945e94e8c22e621dc8542bc34701ecf8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T12:52:46Z","title_canon_sha256":"a21eacb5302313c91af3f3b33b8bde935783f262fb92b84acc245a27e37e5c1a"},"schema_version":"1.0","source":{"id":"2605.13460","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.13460","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"arxiv_version","alias_value":"2605.13460v1","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.13460","created_at":"2026-05-18T02:44:41Z"},{"alias_kind":"pith_short_12","alias_value":"OWOIBHOUOQMI","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_16","alias_value":"OWOIBHOUOQMIOFET","created_at":"2026-05-18T12:33:37Z"},{"alias_kind":"pith_short_8","alias_value":"OWOIBHOU","created_at":"2026-05-18T12:33:37Z"}],"graph_snapshots":[{"event_id":"sha256:450818438555ed1c7048c82643b638279c225dcfe6a302f25e992b889dc21117","target":"graph","created_at":"2026-05-18T02:44:41Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":4,"items":[{"attestation":"unclaimed","claim_id":"C1","kind":"strongest_claim","source":"verdict.strongest_claim","status":"machine_extracted","text":"we show that F_p(x):=x^{2p}+2x^{p}+2 is monogenic if and only if p is not a Wieferich prime."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The standard definitions of monogenic polynomials and Wieferich primes are assumed to apply directly, along with the irreducibility of F_p(x) over Q for prime p (details of the proof not visible in abstract)."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich."}],"snapshot_sha256":"c7af62e100f6350fb28103698cee064b51eb445d9999705769c82a839a76ea4e"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"A prime $p$ is called a Wieferich prime if $2^{p-1}\\equiv 1 \\pmod{p^2}$.\n  A monic polynomial $f(x)\\in {\\mathbb Z}[x]$ of degree $N\\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\\mathbb Q}$ and\n  $\\{1,\\theta,\\theta^2,\\ldots,\\theta^{N-1}\\}$\n  is a basis for the ring of integers of ${\\mathbb Q}(\\theta)$, where $f(\\theta)=0$. In this article, we show that ${\\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime.","authors_text":"Lenny Jones","cross_cats":[],"headline":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T12:52:46Z","title":"Wieferich Primes and Monogenic Trinomials"},"references":{"count":72,"internal_anchors":0,"resolved_work":72,"sample":[{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":1,"title":"Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag , 2000","work_id":"18906986-5d90-4d2f-a55e-c24246f72c28","year":2000},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":2,"title":"J. Harrington and L. Jones, A note on generalised Wall-Sun-Sun primes , Bull. Aust. Math. Soc. 108 , No. 3, 373--378 (2023)","work_id":"4884c648-debc-4f3a-ba36-c0a9728a08ca","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":3,"title":"A. Jakhar, S. Khanduja and N. Sangwan, Characterization of primes dividing the index of a trinomial, Int. J. Number Theory 13 (2017), no. 10, 2505--2514","work_id":"ca49dad4-133f-4833-8f9d-82da90952231","year":2017},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":4,"title":"Jones, Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials , Albanian J","work_id":"4ce457e6-76f4-43bc-be0a-4743028f2e85","year":2023},{"cited_arxiv_id":"","doi":"","is_internal_anchor":false,"ref_index":5,"title":"Jones, A new condition for k -Wall-Sun-Sun primes , Taiwanese J","work_id":"2359ac34-1500-4526-8b94-26d0434b552b","year":2024}],"snapshot_sha256":"33af78b26a99569bf50db5317d09e8a881051f307fc75ae57311abc6f4027d5f"},"source":{"id":"2605.13460","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:19:53.597455Z","id":"611ec171-100e-47b9-8f6b-c9c6730fe4be","model_set":{"reader":"grok-4.3"},"one_line_summary":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich.","strongest_claim":"we show that F_p(x):=x^{2p}+2x^{p}+2 is monogenic if and only if p is not a Wieferich prime.","weakest_assumption":"The standard definitions of monogenic polynomials and Wieferich primes are assumed to apply directly, along with the irreducibility of F_p(x) over Q for prime p (details of the proof not visible in abstract)."}},"verdict_id":"611ec171-100e-47b9-8f6b-c9c6730fe4be"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:330531c1cf9dd9e7b8a4a2c230c4ebc2d68c0b70678d3f3e4d870140bb2791e4","target":"record","created_at":"2026-05-18T02:44:41Z","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":"b4eddb356a8a5d555717ca09da4945e4945e94e8c22e621dc8542bc34701ecf8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T12:52:46Z","title_canon_sha256":"a21eacb5302313c91af3f3b33b8bde935783f262fb92b84acc245a27e37e5c1a"},"schema_version":"1.0","source":{"id":"2605.13460","kind":"arxiv","version":1}},"canonical_sha256":"759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4","first_computed_at":"2026-05-18T02:44:41.718615Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:41.718615Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qEVWQMMrKoiSVqXFiTsbEVwXs5SiMGU3GyitljcHUg890fhUHmAtY968lQlIaeDU/QeHiBIICeYSp0hBPmKmBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:41.719081Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.13460","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:330531c1cf9dd9e7b8a4a2c230c4ebc2d68c0b70678d3f3e4d870140bb2791e4","sha256:450818438555ed1c7048c82643b638279c225dcfe6a302f25e992b889dc21117"],"state_sha256":"2d5f604b652e6312731521f05a954eea571202f28f7ae45fdda1bea17ef564ce"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y16RDhzD2sptvvvvBm3XDzlR8tgfc2Gkct4+XdpShQLMexasaGfZNmnJtXIeD0CWDGQFVaK7JpJykSutxs0KBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T15:34:52.128671Z","bundle_sha256":"526318590f8a68fd8f25ab173732ac942b164c123c6f9f11f710aba8560aac07"}}