Sum of Distinct Biquadratic Residues Modulo Primes

TL;DR

This paper proves two conjectures about the sum of outputs of a specific biquadratic polynomial modulo primes, providing a detailed formula depending on prime residue classes and applying it to classify sums over all integer coefficients.

Contribution

It establishes a formula for the sum of distinct polynomial outputs modulo primes and classifies these sums in terms of quadratic and fourth powers, extending previous conjectures.

Findings

Proved formulas for sums of polynomial outputs modulo primes.

Classified sums based on quadratic and fourth power residues.

Extended results to non-monic polynomials.

Abstract

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo any prime $p\ge 7$. Here, $c$ is an integer not divisible by $p$ and $e$ is any integer. The formula splits into eight cases, depending on the remainder of $p$ modulo $8$ and whether $c$ is a quadratic residue modulo $p$. The formula quickly extends to the non-monic case. We then apply the formula to prove a classification of the set of such sums in terms of the sets of squares and fourth powers, when $c$ in $x^4 + cx^2$ is varied over all integers with a fixed prime modulus $p\ge 7$. The sum and the set of sums are manually computed for the excluded prime moduli $p=3,5$.