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An identity-only residue set in Kohen’s Conjecture 1

Abstract

The unrestricted generalized central trinomial residue generation conjecture fails at p=3, a=3, b=1.

Nadav Kohen, Density and Symmetry in the Generalized Motzkin Numbers mod p, arXiv:2411.03681v2, Conjecture 1, asks whether the first p generalized central trinomial coefficients generate the multiplicative group of the prime field whenever none is divisible by p. The conjecture quantifies over all integer parameters a and b. Its defining constant term is taken in the integer Laurent polynomial ring. The source is https://arxiv.org/html/2411.03681v2.

Definition 1.1 (The defining Laurent polynomial).

Formalization. D5/S3/ArithUnits/KohenMotzkinGenerationRefutation.step (✓ std3).

Source. Repository-derived.

Commentary.

The integer parameters a and b give the Laurent polynomial aX^(-1)+b+aX. LaurentPolynomial.C includes an integer as a constant, while LaurentPolynomial.T takes an integer exponent, so the negative exponent is literal. No restriction on either parameter is imposed.

Definition 1.2 (Integer constant terms).

Formalization. D5/S3/ArithUnits/KohenMotzkinGenerationRefutation.central (✓ std3).

Source. Repository-derived.

Commentary.

For every natural index n, central(a,b,n) is the coefficient of exponent zero in step(a,b)^n. This is the source’s generalized central trinomial coefficient, computed as an integer before taking any residue. The generalized Motzkin coefficients are a different sequence and are not substituted here.

Definition 1.3 (The first p residues in the unit group).

Formalization. D5/S3/ArithUnits/KohenMotzkinGenerationRefutation.generators (✓ std3).

Source. Repository-derived.

Commentary.

A unit u belongs to generators(p,a,b) exactly when its underlying residue is central(a,b,n) modulo p for some natural n<p. The indices 0 through p-1 are the canonical representatives of the source’s indices in the prime field. Under the nondivisibility premise every such residue is nonzero and hence a unit. This set therefore retains all the source generators.

Definition 1.4 (The complete unrestricted generation assertion).

Formalization. D5/S3/ArithUnits/KohenMotzkinGenerationRefutation.claim (✓ std3).

Source. Repository-derived.

Commentary.

For every natural prime p and every pair of integers a,b, assume that p divides none of central(a,b,n) for 0<=n<p. The conclusion says that the subgroup generated by their residues is the whole multiplicative unit group of ZMod p. Subgroup.closure includes the identity, inverses and arbitrary finite products. The premise adds no condition that p does not divide a or b. This is the generating assertion of Conjecture 1; the subsequent statements introduced as equivalent or consequent in the source do not replace it.

Theorem 1.5 (The assertion is false).

Proof. Machine-checked in Lean as D5/S3/ArithUnits/KohenMotzkinGenerationRefutation.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

Specialize the universal assertion to p=3, a=3 and b=1. Extracting the constant terms of the zeroth, first and second powers gives 1, 1 and 1+2*3^2=19. These are all the indices less than three, and none of the three integers is divisible by three. Their residues are all one. Thus every generator belongs to the identity subgroup, and the least-subgroup property of Subgroup.closure places the whole generated subgroup there. The residue two is an actual unit modulo three and differs from one. It belongs to the whole unit group but cannot belong to the generated subgroup, contradicting the asserted equality. The coefficient extraction uses distributivity and exact integer arithmetic; the subgroup obstruction uses the group closure law. The proof refutes the unrestricted assertion. A variant imposing that p does not divide a is a different assertion and is not settled here.

References