A generalization of the Goresky-Klapper conjecture, Part I

TL;DR

The paper proves that for large primes, certain polynomial permutations cannot map entire residue classes mod n to a single class, except for specific well-understood cases, extending the Goresky-Klapper conjecture.

Contribution

It generalizes the Goresky-Klapper conjecture by characterizing permutation maps of residue classes mod n for large primes, identifying when such mappings are only possible in known special cases.

Findings

Permutation maps of the form $A x^k$ cannot collapse residue classes mod n for large p.

The only exceptions are specific linear and quadratic maps depending on the parity of n.

The result extends the understanding of residue class mappings in modular arithmetic.

Abstract

For a fixed integer $n\geq 2,$ we show that a permutation of the least residues mod $p$ of the form $f(x)=Ax^k$ mod $p$ cannot map a residue class mod $n$ to just one residue class mod $n$ once $p$ is sufficiently large, other than the maps $f(x)=\pm x$ mod $p$ when $n$ is even and $f(x)=\pm x$ or $\pm x^{(p+1)/2}$ mod $p$ when $n$ is odd.