bibkey: niedbala2026carlitz authors: David Niedbala Giraudin year: 2026 title: “A counterexample to a conjecture of Thakur on Carlitz-Wieferich primes” doi: 10.48550/arXiv.2607.15305 url: https://arxiv.org/abs/2607.15305 claim: “Version 2, Conjecture 4.2: over F_(19^3), gcd(T^(q^5)-T,M_(5,q)) equals mu(T^q-T), with mu=X^5+5X^3+3X^2-4X-9.” strata_touched:

• D5/S3/Arith/FunctionField/CarlitzFiveOrbit license: CC-BY-4.0 triage: anchor

Carlitz degree-five exactness

The relevant primary version is arXiv:2607.15305v2, dated 22 July 2026: https://arxiv.org/html/2607.15305v2 . The abstract and version comment explicitly say that the former exactness theorem was corrected to a conjecture. It is not valid to quote the superseded version-one exactness label as a proof.

Verified locator

• DOI: 10.48550/arXiv.2607.15305
• URL: https://arxiv.org/abs/2607.15305

Theorem 1.1 supplies an explicit irreducible Carlitz-Wieferich quintic over F_(19^3). Theorem 4.1 proves that its 6859 translates have product mu(T^q-T) and that this product divides the gcd. Those results are established inputs of the source, not the new target. Conjecture 4.2 asks whether any additional factor exists. The discussion isolates a possible difference eta=theta^q-theta of degree fifteen over F19; the low-degree difference case was already accounted for.

The companion Effective determination of Carlitz-Wieferich primes of given degree is listed as in preparation in version 2. Searches for the exact arXiv identifier, exactness conjecture number, companion title, and combined Carlitz-Wieferich completeness terms did not locate a later published proof in the checked sources. This is a bounded literature check, not an assertion of worldwide priority or author/editor acceptance.

The new certificate uses the literal nested residual from equation (2), its five conjugates, and the literal mu from Conjecture 4.2. It proves that all solutions of the resulting orbit equations have quintic difference. The corresponding ordinary proof concludes exactness and also handles all constant fields F_(19^s). The original counterexample to Thakur’s suggestion is not counted again.

Earlier primary sources for the criterion are:

• D. S. Thakur, Fermat versus Wilson congruences, arithmetic derivatives and zeta values, Finite Fields and Their Applications 32 (2015), 192-206.
• A. S. Bamunoba and J. Bergstrom, A search for c-Wieferich primes, International Journal of Number Theory 17 (2021), 1599-1616, https://arxiv.org/abs/2011.11727 .

Function-field Carlitz-Wieferich primes are distinct from integer Wall-Sun-Sun primes. The characteristic-nineteen elimination certificate has no asserted implication of integer WSS existence.

Unified owner and the all-characteristic continuation

All CF1-CF5 proofs now live under the existing owner Problems/wall-sun-sun-golden-unit-lift.md, followed by CX1-CX6. The former separate Carlitz problem file was removed only after its complete mathematics was retained there. This Library note remains a source record, not a second open-problem entry. The existing CarlitzFiveOrbit Lean/Scribe pair is unchanged.

The source’s Proposition 5.2 reports finite prime-field computations, with its degree-five row covering p=3,7,11,13,17,19,23,29,31,37. It does not give the uniform all-odd-characteristic classification developed in CX. The new integer five-orbit certificate restricts odd characteristic to5,19,263 and 519555805809266011. Additional exact certificates and constructions give extension classes s=3 modulo5 for263 and s=4 modulo5 for the large prime. Together with CF this excludes degree-five nonconforming examples over every odd prime field. These new mathematical conclusions are not attributed to the source author, and do not answer the source’s prime-field question in higher degrees.

The companion paper remains listed as in preparation in the checked v2. Exact searches for Carlitz-Wieferich together with263, the large characteristic, the companion title and degree-five completeness did not locate the new families in the checked primary sources. The known characteristic19 result and its already delivered completeness proof are not counted again. Neither absence from indexed search nor an unpublished companion’s unknown contents establishes worldwide priority. External acceptance remains unconfirmed.

Applicability audit: the characteristic-selection construction fails

This records a limitation of the proposed transfer inside the same Wieferich problem family. It is not a new open-problem entry or an attribution to the source author. The attempted construction takes the rational characteristic p>5 of a degree-five Carlitz-Wieferich example over some F_(p^s), and uses that same p as an integer Fibonacci WSS candidate.

The existing CX characteristic restriction leaves only

$$p=19,263,P_{\ast }=519555805809266011.$$

Exact modular Fibonacci evaluation gives, with epsilon=(5/p),

$$\begin{array}{crr}p& F_{p-ϵ}\mathrm{mod}{p}^2& q_p=F_{p-ϵ}/p\mathrm{mod}p\\ 19& 57& 3\\ 263& 27615& 105\\ P_{\ast }& 218822702976076845315377336456939760& 421172664282398160.\end{array}$$

Every quotient is nonzero. The primality of P_* has a recursive Lucas certificate. Thus the complete characteristic-selection route produces no integer WSS prime. This does not exclude other maps between the two arithmetic problems, or invalidate the function-field constructions.

The unchanged five residuals cannot lift to characteristic p squared

Write R(a,b,c,d)=1-d(1-c(1-b(1-a))) and tau(a,b,c,d)=(b-a,c-a,d-a,-a). The actual source CarlitzFiveCharacteristic.lean contains the integer identity

$$\sum _{i=0}^{4}S({\tau }^i(a,b,c,d))R({\tau }^i(a,b,c,d))=D,$$

where

$$D=4673196650932024062540600=3^2\cdot 8\cdot 25\cdot 19\cdot 263\cdot P_{\ast }.$$

Its polynomial identity uses only integer coefficients and ring operations, even though the public source theorem is stated for fields. Therefore it is valid in every commutative ring. If all five residuals vanish in a ring of exact characteristic p^e, then p^e divides D. For p in {19,263,P_*}, the exponent of p in D is exactly one. Consequently there is no such simultaneous solution in any ring of exact characteristic p^e with e>=2. In particular, none of the known residual solutions lifts to an unramified length-two ring while preserving all five equations. The case p=5 is not covered by this first-power obstruction.

This is a statement about the rational constant p squared. The Carlitz congruence modulo a polynomial P(T)^2 is in equal characteristic p, where p itself is zero. There is no contradiction between the two statements.

A concrete first-order obstruction and the unique scalar deformation

Fix p=19 and the monic integer lift

$$\tilde{P}(T)=T^5+13T^3+3T^2+10T+15.$$

In W=(Z/19^2)[t]/(Ptilde), let sigma be the unramified automorphism lifting the 19^3-power Frobenius. In the basis (1,t,t^2,t^3,t^4), its value at t is (176,41,237,82,105) modulo361. Polynomial substitution verifies Ptilde(sigma(t))=0 and sigma^5(t)=t. Put

$$\mathcal{E}(x)=R(\sigma x-x,{\sigma }^{2}x-x,{\sigma }^{3}x-x,{\sigma }^{4}x-x).$$

For every z in F_(19^5), exact first-order expansion gives

$$\mathcal{E}(t+19z)/19=e+Jz\text{in }{\mathbb{F}}_{19}^5,$$

where

$$e=\left(\begin{array}{c}10\\ 8\\ 1\\ 3\\ 11\end{array}\right),J=\left(\begin{array}{ccccc}0& 5& 10& 16& 13\\ 0& 6& 11& 10& 0\\ 0& 14& 5& 0& 5\\ 0& 0& 14& 2& 2\\ 0& 13& 16& 17& 3\end{array}\right).$$

The row w=(4,2,14,15,0) satisfies wJ=0 and we=1. Hence no choice of z can make the original residual vanish modulo361. The four nonconstant columns have rank four; the minor obtained by deleting the first row has determinant8 modulo19. Thus the attainable error vectors form exactly the affine hyperplane w y=1, with19 preimages for each error. Constant translations are precisely the one-dimensional kernel. This conclusion concerns every lift, not a finite sample of z values.

If one changes the equation to E(x)=19c with c in F19, solvability instead requires 4c=1, hence c=5. Fixing the constant coefficient of z to zero gives one solution z=(0,13,0,18,9). The augmented derivative in the four nonconstant coordinates and c has nonzero determinant11. Successively correcting the five coordinates therefore gives, in the unramified degree-five extension of Z_19, a unique normalized x and scalar kappa in19 Z_19 such that E(x)=kappa, x=t modulo19. Its first digit is kappa/19=5 modulo19, so v_19(kappa)=1. The same sigma makes all five translated-origin residuals exactly equal to kappa. To precision19^4 the constructed scalar is80237. The inverse derivative proves existence and uniqueness at every precision: at each step the same invertible residual linear map determines the next five digits, and completeness supplies their compatible limit.

The corresponding exact calculations for263 and P_* also give rank-four linearization and a nonzero augmented determinant. With coefficient lifts in [0,p-1] of the canonical polynomials P_263 and P_P* from CX, their unique first scalar digits are respectively248 and311747520347328037. The augmented determinants are17 and140776401079060105. The constructed scalars have p-adic valuation one in both cases. These scalar digits depend on the stated integral models and normalization; they are not asserted to be invariant under arbitrary changes to the residual equation.

This construction solves a deformed equation E=kappa with kappa nonzero. It is not a mixed-characteristic solution of E=0 and not an integer WSS construction. Merely allowing a correction parameter can hide the original obstruction, so the parameter must remain part of the conclusion.

Comparison with the fixed golden lift, using the current owner CG.2/CG.6

For p>5 retain the actual golden ring, phi^2=phi+1, d=2phi-1, d^2=5, and sigma equal to the identity or conjugation according to epsilon=(5/p). For a lift x=phi+pz define, by division before reduction,

$$U(x)=(x^2-x-1)/p,V(x)=(x^p-\sigma x)/p\mathrm{mod}p.$$

The existing owner formula for the golden p-derivation gives

$${\Delta }_p=({\varphi }^p-\sigma \varphi )/p=\frac{5+ϵd}{4}{q}_p(\mathrm{mod}p).$$

Expansion in the actual quadratic ring yields

$$U(x)=dz,V(x)={\Delta }_p-\sigma z,V(x)+\sigma (d^{-1}U(x))={\Delta }_p.$$

The coefficient (5+epsilond)/4 is a unit: its norm is5/4. Thus simultaneous preservation of the original quadratic equation and the Frobenius equation is equivalent to q_p=0. If only the Frobenius equation is kept, the unique correction is z=q_p(phi+2)/2. It preserves norm minus one but changes the trace to 1+5p q_p/2 modulo p^2; the fixed quadratic then has residual p(5q_p/2)*phi. It solves the modified trace-parameter quadratic, not the original one unless q_p=0.

This comparison is an explicit consequence of the already recorded CG identities. It is not a newly independent constraint on the WSS zero set. The useful new boundary is that the five-residual characteristic restriction cannot simply be carried through a mixed-characteristic lift. A successful transfer must supply the genuine integer lifting equations and preserve their fixed parameters, rather than infer integer WSS from an adjustable residual or from the existence of some lifted root. Neither a WSS prime nor an unbounded WSS exclusion family is proved by this audit.