New empty cells in the Carlitz–Wieferich grid, and a correction to a published table
Abstract
Let p be an odd prime, q=pᵉ, A=𝔽_q[T], and let ρ be the Carlitz module. A monic prime P ∈ A is a c-Wieferich prime (to base 1) when the Carlitz–Fermat congruence lifts, ρ_P(1) ≡ 1 pmodP². Thakur excluded degrees 2 and 3 in odd characteristic and suggested that the degree of a c-Wieferich prime is always divisible by p; the suggestion was refuted in 2026 by an explicit degree-5 prime over 𝔽₁₉³, and the paper that refuted it published a grid of thirty pairs (d,p) with p ∤ d for which 𝔽ₚ[T] was shown, by exhaustive computation, to contain no c-Wieferich prime of degree d. We extend that grid by thirty-one cells in two directions. The prime-field rows are continued past their published frontier: degree 5 is now settled for every odd p ≤ 101 with p ≠ 5, degree 6 for every odd p ≤ 37 with p ≠ 3, degree 7 for p ≤ 19, degree 8 for p ≤ 13, and degrees 9 and 10 at p=7. A second axis, absent from the published table, is opened: the cells (d,q) with q a proper prime power, namely (5,9), (5,27), (5,49), (5,81), (6,9), (6,25), (7,9) and (8,9), of which (5,27) is the smallest cubic-extension analogue of the cell in which the counterexample lives. All thirty-one are empty; together the sweeps run over 843 189 705 cosets. We also correct a published table: the entry T⁵+T+1 recorded as the c-Wieferich prime of least degree in 𝔽₅[T] is reducible, the intended polynomial being T⁵+4T+1. Under the natural q⁻ᵈ heuristic the published grid was a 0.94 the new cells multiply the numerical evidence for Thakur's phenomenon by about four. All statements are machine-checked in Lean 4.
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Source snapshot 2026-08-30 15:34 UTC
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Claim ledger
Stated results
CW1routine2026-08-28
The field model is proved, not assumed: for every cell, ctxOK certifies that Fₚ[z]/(h) is a field of order pᵉᵈ (criterion: the p-power matrix fr satisfies frⁿ = I and rank(fr - I) = n - 1), that phi = frᵉ is the q-power map, and that the sweep's index set is a transversal of the cosets of F_q; ctxOK rejects reducible moduli, including a sextic whose two cubic factors are irreducible, and the answers do not depend on which irreducible modulus is used
CW2known data2026-08-28
Every c-Wieferich prime known over an odd prime field, certified: T⁵+4T+1 over F₅, T⁶+T⁴+T³+T²+2T+2, T⁹+T⁶+T⁴+T²+2T+2, T¹2+2T¹0+T⁹+2T⁴+2T³+T²+1 and T¹5+T¹3+T¹2+T¹1+2T¹0+2T⁷+2T⁵+2T⁴+T³+T²+T+1 over F₃, T¹0+3T⁶+4T⁵+T²+T+1 over F₅, T⁷+6T+3 over F₇; the exhaustive coset counts of the nonempty cells (5,5), (6,3), (7,7), (9,3), (12,3), (15,3), (10,5), (3,9), (5,25), with the minimal polynomial recovered from the sweep at (5,5), (6,3), (7,7); fourteen entries of the count table of arXiv:2011.11727 through the identity cosets = s*|B_(pᵐ,s)|; and all eleven published terms of OEIS A271781
CW3known data2026-08-28
All thirty cells of Proposition 5.2 of arXiv:2607.15305v2 re-derived: no c-Wieferich prime of degree d in Fₚ[T] for d=5 and p in 3,7,11,13,17,19,23,29,31,37; d=6 and p in 5,7,11,13,17,19,23; d=7 and p in 3,5,11,13; d=8 and p in 3,5,7; d=9, p=5; d=10, p=3; d=11, p in 3,5; d=13, p=3; d=14, p=3
CW4known2026-08-28
Theorem 1.1 of arXiv:2607.15305v2 certified in Lean: P(T) = T⁵ + (11+17c+9c²)T⁴ + (3+7c+18c²)T³ + (2+5c+6c²)T² + (3+3c+11c²)T + (6+17c+5c²) over F_(19³) = F₁9[c], c³ = 8c²+4c+11, is an irreducible c-Wieferich prime of degree 5, and 19 does not divide 5
CW5known2026-08-28
No c-Wieferich prime of degree 2, 3 or 4 over Fₚ for p in 3,5,7,11,13, none of degree 4 over F₉, F₂5, F₂7, F₄9, F₁21, and none of degree 2 or 3 over F₂5, F₄9, F₁21 or of degree 2 over F₉
CW6candidate2026-08-28
The d = 5 row of Proposition 5.2 continued: no c-Wieferich prime of degree 5 in Fₚ[T] for p = 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101 – with the published row this makes degree 5 settled for every odd p <= 101 with p different from 5
CW7candidate2026-08-28
The d = 6 row of Proposition 5.2 continued: no c-Wieferich prime of degree 6 in Fₚ[T] for p = 29, 31, 37 – with the published row this makes degree 6 settled for every odd p <= 37 with p different from 3
CW8candidate2026-08-28
The d = 7 row of Proposition 5.2 continued: no c-Wieferich prime of degree 7 in Fₚ[T] for p = 17 and p = 19 – with the published row this makes degree 7 settled for every odd p <= 19 with p different from 7 (and p = 7 is not empty: T⁷+6T+3)
CW9candidate2026-08-28
The d = 8 and d = 9 rows of Proposition 5.2 continued: no c-Wieferich prime of degree 8 in F₁1[T] or F₁3[T], and none of degree 9 in F₇[T]
CW10candidate2026-08-28
The d = 10 row of Proposition 5.2 continued: no c-Wieferich prime of degree 10 in F₇[T]. The published row has p = 3 only, and p = 5 is the p | d case and is not empty
CW11candidate2026-08-28
A second axis the published grid does not have: cells over non-prime F_q. No c-Wieferich prime of degree 5 in F₉[T], F₂7[T], F₄9[T] or F₈1[T]; none of degree 6 in F₉[T] or F₂5[T]; none of degree 7 or 8 in F₉[T]
CW12routine2026-08-28
Negative controls: a too-large claim ('every cell is empty') refuted by five nonempty cells with their exact coset counts, a too-small claim ('there is a c-Wieferich prime of degree 2, 3 or 4 over a small field') refuted, and the degree hypothesis of Lemma 2.7 shown non-vacuous – dropping it makes the empty cell (6, F₉) report 2 witnesses, and changes (12, F₃) from 4 to 6 and (10, F₅) from 2 to 3
CW13correction2026-08-28
A published table entry is wrong: the F₅ row of Table 1 of arXiv:2011.11727 names T⁵ + T + 1 as the c-Wieferich prime of least degree in F₅[T], but that polynomial is reducible (T = 2 is a root). The polynomial is T⁵ + 4T + 1, which is irreducible and c-Wieferich
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- Korea Superintelligence Labs
- Source context
- Let p be an odd prime, q = pᵉ, and A = F_q[T]. Write [n] = T^(qⁿ) − T, and let ρ be the Carlitz module: the F_q-algebra map A → Aτ into the twisted polynomial ring (τ x = x^q τ) determined by ρ_T = T + τ. For every monic prime P ∈ A one has ρ_P(1) ≡ 1 (mod P) — the Carlitz analogue of Fermat's little theorem. P is a c-Wieferich prime (to base 1) when the congruence lifts:
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- 2026-09-07 03:53 UTC
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