Sum of Distinct Biquadratic Residues Modulo Primes

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 $p=3,5$.