A generalization of the Goresky-Klapper conjecture, Part II

TL;DR

This paper investigates the distribution of residues in permutation maps of the form f(x)=Ax^k mod p, showing that except for specific cases, the image of residue classes covers all classes for large primes, with detailed exceptions analyzed.

Contribution

It generalizes the Goresky-Klapper conjecture by analyzing the residue class coverage of permutation maps, identifying conditions under which certain classes are missed, and providing explicit counts of such cases.

Findings

For fixed n≥2, the image of each residue class mod n contains elements from all classes for large p.

Exact counts of A values where some residue class misses others are provided, especially for linear and quadratic maps.

Most permutation maps cover all residue classes, with explicit exceptions characterized.

Abstract

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements from every residue classe mod $n$, once $p$ is sufficiently large. If $f(x)=Ax$ mod $p$, then for each $p$ and $n$ there will be exactly $(1+o(1))\frac{6}{\pi^2}n^2$ readily describable values of $A$ for which the image of some residue class mod $n$ misses at least one residue class mod $n,$ even when $p$ is large relative to $n$. A similar situation holds for $f(x)=Ax^{(p+1)/2}$ mod $p$.