On permutations derived from integer powers $x^n$

TL;DR

This paper characterizes how the base-$p$ representations of integers relate to their powers, establishing bijections and explicit formulas for permutations within residue classes mod $p^{l+1}$, with applications to encoding integers.

Contribution

It introduces a general theorem linking base-$p$ representations of integers and their powers, providing explicit formulas for permutations across residue classes, including zero residue classes.

Findings

Established bijective mappings between integers and their powers in residue classes

Derived explicit formulas for coding shifts depending on $n$ and $p$

Extended results to include zero residue class mappings for all positive integers

Abstract

We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the existence of bijective mappings (permutations) between all $p^l$ members $x$ of each non-zero residue class mod $p$ satisfying $x < p^{l+1}$. These mappings are obtained as the integer part of ${x^p}{p^{-\alpha}}$ for a particular positive integer $\alpha$, depending on $n$ and $p$, called the "coding shift", for which an explicit formula is given. For relatively prime $n$ and $p$, $\alpha = 1$ and the result follows directly from properties of the multiplicative group of invertible elements modulo $p^{l+1}$. We extend our result for general $n$ also to identify the coding shift required to obtain such bijective mappings for members of the zero residue class mod $p$, demonstrating that such bijective mappings (or encodings) can be found for any finite $l$ and for all positive integers $x < p^{l+1}$.