bibkey: sharma2026primitivepolynomials authors: Avnish K. Sharma year: 2026 title: “Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures” doi: null url: https://arxiv.org/abs/2608.07262v2 claim: “Conjecture 4.2 asserts that for every odd prime p, every finite field K of cardinality p^2, and every multiplicative generator λ of K, X^p+X+λ is the monic minimal polynomial of a multiplicative generator in a degree-p extension.” strata_touched:

• D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation license: CC-BY-4.0 triage: anchor

Primitive polynomials of the form g(x)+λ

Avnish K. Sharma’s arXiv version 2 (arXiv:2608.07262v2, August 30, 2026) studies primitive polynomials of the form (g(x)+\lambda) over finite fields. Conjecture 4.2, printed page 16, states (mathematical typography transliterated):

Let p be an odd prime. Then, for every primitive element λ∈F_(p²), the polynomial x^p+x+λ is primitive over F_(p²).

Its full scope is: for every odd prime (p), every finite field (K) of cardinality (p^2) and characteristic (p), and every (\lambda\in K) satisfying the source multiplicative-generator condition (\operatorname{ord}(\lambda)=|K|-1), the polynomial (X^p+X+\lambda) is the monic minimal polynomial of a multiplicative generator in a degree-(p) extension. Here “primitive” means the source’s multiplicative-generator condition, not Polynomial.IsPrimitive (coefficient content).

Verified locator

Versioned arXiv record: https://arxiv.org/abs/2608.07262v2. The primary sources are the versioned PDF and versioned HTML. The audited PDF SHA256 is 6b4318dbf8743a53ee8541993bd2a63332828fcd7632937861e2e656aab2d635; the HTML SHA256 is 494fb73698fbf2bf9c09b38656be21f304640ef95a0da8b58bf7dcb7975b12d1. Remark 4.2 checks every primitive lambda only for primes 3 through 37; 41 is outside that reported range. The existential Conjecture 4.1 is a different statement and is not refuted here.

Formal source and refutation

The Lean module D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation.lean defines the source predicate SourcePrimitivePolynomial, the primitive-lambda predicate, and the exact coefficient-parameterized polynomial. Its public fullClaim is the complete universal Conjecture 4.2 with the source multiplicative order and minimal-polynomial meaning. The public result has type Not (claimFor SourceLeadingCoefficient); it is the unconditional negation of that full claim.

The certificate uses [ K = \operatorname{QuadraticAlgebra}(\mathbb Z/41\mathbb Z,3,0),\qquad \lambda = 5+u, ] where (u^2=3). The kernel checks that (|K|=41^2=1681), that λ has order (1680=|K|-1). The actual source polynomial is (X^{41}+X+\lambda). The 41 coefficients in certificateCoefficients are multiplied by eight exact product identities. Horner semantics then give, for every field extension (L/K) and every root α of (X^{41}+X+\lambda), [ z = \operatorname{aeval}_{\alpha}(\texttt{certificatePolynomial}), \qquad z^{83}=\alpha. ] If Conjecture 4.2 supplied a degree-41 extension and a primitive root α, then (|L|=1681^{41}). The finite-field exponent for z forces (\alpha^{(1681^{41}-1)/83}=1), while primitivity requires order (1681^{41}-1). Since the exponent is positive and strictly smaller, this is impossible. The argument uses the source’s full universal root certificate and multiplicative-order conclusion; it does not replace them by a finite enumeration or by a coefficient-content statement.

Registration boundary and bounded literature scope

The D5 module contains the mathematical source and refutation, with no inline information registration or judge imports. Registration belongs to the separate Reg package under #5214; this artifact supplies no replacement registration. The complete source claim, primitive-lambda witness and universal root certificate remain in D5. The existing result Describe now binds D5/S3/ArithUnits/SharmaPrimitivePolynomialRefutation.result to Problems/sharma-primitive-polynomial-conjecture-4-2 through a typed OpenProblemResolutionClaim with kind Refuted, following canonical Freeze. The generated Blueprint projects that binding.

The candidate was numerically discovered before preregistration #8851, which records the target, source audit and library search. No first-discovery or publication-priority claim is made. That bounded audit found arXiv v1 dated August 7 and current v2 dated August 30, with no later version found. Checked arXiv/title/author, OpenAlex and Crossref results contained no inspected resolution; Semantic Scholar was rate-limited. This does not establish worldwide absence of a resolution.

Local semantic surfaces and all-ref exact identifier/author history had no exact delivery. All-state GitHub exact/formula searches and a paginated issue/PR-body scan found no exact owner; every issue comment was not searched. The pinned Mathlib search (commit db584cd6d46c92f209a44c0f1c829460d327499d) supplied the quadratic algebra, minimal-polynomial, finite-field cardinality and multiplicative-order APIs. The bounded external Lean search found unrelated GF8/GF256 models and antiderivative uses of the name PrimitivePolynomial, without a proof of this conjecture. The source certificate is repository-derived; no external proof code is transplanted.