On the polynomial values represented by quadratic forms

Abstract

Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate quadratic form $F$ and a univariate polynomial $P$ with integer coefficients, $P(x)$ can be represented by $F$ for infinitely many values of $x$. We develop a method for answering this question for certain cubic and quartic polynomials $P$, as well as for certain polynomials of the form $P(x)=R(Q(x))$, where $R(t)$ and $Q(x)$ are polynomials of degree $3$ and $2$, respectively. Applying this method with $F(y,z)=y^2+z^2$, $R(t)=t^3-4$ and $Q(x)=x^2$, we conclude that $x^6-4$ is a sum of two squares infinitely often. In turn, this implies that the equation $y^2+x^3y+z^2+1=0$ has infinitely many integer solutions. Prior to this work, it was the shortest equation for which it was open whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations whose finiteness problem remains open. All main results of this paper has been formalized in Lean using Aristotle.

Disclosure

“r emerged from extended, multi-round inter- actions between the authors and ChatGPT 5.5 Pro. As is often the case in collaborative work, it is difficult to separate the individual contributions precisely. For example, in one of the rounds, ChatGPT produced the first essentially correct proof that equation (2) has infinitely many integer solutions. That proof relied on substantial hints from the authors; at the same time, those hints were themselves influenced by the model’s response”

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Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-24T00:34:09.244Z

Structural counts