{"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"}