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hexagon:2610.00569v1

math.NT — Number Theory

Polynomial parametrizations for two classes of Diophantine equations

Contributed by Bogdan Grechuk

We prove that the integer solution sets of two classes of Diophantine equations are finite unions of polynomial families with integer coefficients and unrestricted integer parameters. In the first class, each monomial, with at most one exception, contains a variable that occurs to the first power and in no other monomial. The second class consists of equations M=f(t)M=f(t), where MM is a monomial independent of tt with a variable occurring to the first power, and f∈Z[t]f\in\Z[t] splits completely over $\Q$. If MM involves at most two variables, ff may instead be an arbitrary quadratic polynomial. We also give complete formulas for five equations listed as open in the monograph \emph{Polynomial Diophantine Equations: A Systematic Approach} (2024).

Primarily AI-generated textHuman understanding: some partsDiophantine equations; Polynomial parametrization; Integer solutions.
Oct 27, 2026PDF ↗View work →
hexagon:2610.00531v1

math.NT — Number Theory

Long solutions to small equations

Contributed by Bogdan Grechuk, Eugene Go, Avraham Eisenberg

We give complete solutions of thirty-eight Diophantine equations left open in Grechuk's 2024 monograph: fifteen affine equations, five generalized Fermat equations, eight plane quartics, and ten further homogeneous equations. All proofs were generated by AI and require substantial computer algebra computations.

Primarily AI-generated textHuman understanding: some partsDiophantine equations; Chabauty method; Brauer--Manin obstruction
Oct 26, 2026PDF ↗View work →