Pith Number

pith:OWOIBHOU

pith:2026:OWOIBHOUOQMIOFETMDDPPPNFIM

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Wieferich Primes and Monogenic Trinomials

Lenny Jones

The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if the prime p is not Wieferich.

arxiv:2605.13460 v1 · 2026-05-13 · math.NT

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\pithnumber{OWOIBHOUOQMIOFETMDDPPPNFIM}

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

we show that F_p(x):=x^{2p}+2x^{p}+2 is monogenic if and only if p is not a Wieferich prime.

C2weakest 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).

C3one line summary

The trinomial x^{2p} + 2x^p + 2 is monogenic if and only if p is not a Wieferich prime.

References

72 extracted · 72 resolved · 0 Pith anchors

[1] Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag , 2000 2000

[2] J. Harrington and L. Jones, A note on generalised Wall-Sun-Sun primes , Bull. Aust. Math. Soc. 108 , No. 3, 373--378 (2023) 2023

[3] 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 2017

[4] Jones, Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials , Albanian J 2023

[5] Jones, A new condition for k -Wall-Sun-Sun primes , Taiwanese J 2024

Receipt and verification

First computed	2026-05-18T02:44:41.718615Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4

Aliases

arxiv: 2605.13460 · arxiv_version: 2605.13460v1 · doi: 10.48550/arxiv.2605.13460 · pith_short_12: OWOIBHOUOQMI · pith_short_16: OWOIBHOUOQMIOFET · pith_short_8: OWOIBHOU

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/OWOIBHOUOQMIOFETMDDPPPNFIM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 759c809dd4741887149360c6f7bda54308d166c84ed4562c7a21bf89d8ff33a4

Canonical record JSON

{
  "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
  }
}