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Jacobsthal sums and permutations of biquadratic residues

Hai-Liang Wu, Yue-Feng She

TL;DR

This work studies permutation problems induced by 4th-power (biquadratic) residues modulo primes $p$ with $p ≡ 1\ (mod\ 4)$, focusing on the signs of the permutations $τ_p(g)$ and $ρ_p$ built from quadratic/biquadratic residues and primitive roots. It expresses these signs via Jacobsthal sums $φ_k(m)$ and $ψ_k(m)$ and related combinatorial counts, obtaining explicit parity formulas that depend on $p$ modulo $16$ and quartic residuosity data (e.g., $χ_4(2)$) and the quantities $λ_p$, $ε_p$, etc. For $p ≡ 9\ (mod\ 16)$ the sign of $τ_p(g)$ is independent of the primitive root, while for $p ≡ 1\ (mod\ 16)$ a root-dependent refinement appears; additionally, $sgn(ρ_p)=(-1)^{λ_p+ε_p}$. These results connect higher-power reciprocity with permutation statistics in finite fields and utilize cyclotomic-field techniques to relate residue structure to permutation signs.

Abstract

Let $p\equiv1\pmod 4$ be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo $p$.

Jacobsthal sums and permutations of biquadratic residues

TL;DR

This work studies permutation problems induced by 4th-power (biquadratic) residues modulo primes with , focusing on the signs of the permutations and built from quadratic/biquadratic residues and primitive roots. It expresses these signs via Jacobsthal sums and and related combinatorial counts, obtaining explicit parity formulas that depend on modulo and quartic residuosity data (e.g., ) and the quantities , , etc. For the sign of is independent of the primitive root, while for a root-dependent refinement appears; additionally, . These results connect higher-power reciprocity with permutation statistics in finite fields and utilize cyclotomic-field techniques to relate residue structure to permutation signs.

Abstract

Let be a prime. In this paper, with the help of Jacobsthal sums, we study some permutation problems involving biquadratic residues modulo .

Paper Structure

This paper contains 3 sections, 10 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem A']}
  3. Proof of Theorem \ref{['theorem B']}

Key Result

Theorem 1.1

Let $p\equiv1\ ({\rm{mod}}\ 8)$ be a prime of the form $a^2+4b^2$ with $a\equiv-1\ ({\rm{mod}}\ 4)$ and $b>0$. Then we have the following results. (i) If $p\equiv9\ ({\rm{mod}}\ 16)$, then ${\rm sgn}(\tau_p(g))$ is independent on the choice of $g$. And we have (ii) If $p\equiv 1\ ({\rm{mod}}\ 16)$, then we have In this case, if we let then we have

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 8 more