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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."},"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":true,"formal_links_present":false},"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"},"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"},"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. 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