# 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.

## Key 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 $x < p^{l+1}$.

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Source: https://tomesphere.com/paper/1907.01890