Jamal Agbanwa | Searcharxiv

arXiv2026

On the polynomial values represented by binary quadratic forms

Many Diophantine equations can be reduced to the question of whether, for a given non-degenerate integral binary 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 degrees $3$ 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 unknown whether its integer solution set is finite or infinite. We conclude with a list of the new shortest equations for which the finiteness problem remains open. All main results of this paper have been formalized in Lean using Aristotle.

math.GM↗

arXiv2025

A Closed-Form Symbolic Generator: $A^n + B^n = C^n + D^n$, for $n = 2,3$

We present a unified framework for constructing integer solutions to $A^{n} + B^{n} = C^{n} + D^{n}$ for $n=2,3$. For $n=2$, we derive explicit formulas for any solutions via differences of squares. For $n=3$, we introduce general formulas that include the Hardy-Ramanujan number 1729 for instance, we also construct a symbolic generator that produces infinitely many integer solutions to the Diophantine equation A^3 + B^3 = C^3 + D^3 . While the resulting formulas for $A,B,C,D$ from the symbolic generator developed do not span every single number expressible as a sum of two positive cubes in at least two distinct ways, our method provides a closed-form, algebraic parametrization in terms of a single variable, expressing each term as a radical-exponential function of an integer parameter $c_1$. The generator leverages nested radicals and exponents of algebraic numbers, $α, β$ derived from the recurrence structure of the Diophantine constraint. This work represents the first symbolic, recursive generator of its kind and offers a pathway toward approaching higher powers of this problem from a different lens. These methods exploit structural links between binomial expansions and Diophantine constraints, offering a foundation for extensions to higher powers.

math.GM↗