The Carlitz group of the rationals

TL;DR

This paper solves a challenge question about rational maps with square difference quotients and explores a prime-related function generalizing Carlitz's result, combining theoretical and computational methods.

Contribution

It provides a complete solution to a specific functional equation and introduces a new prime-based function generalizing Carlitz's theorem.

Findings

Characterization of all rational maps with square difference quotients

Identification of a key prime function related to the problem

Extension of Carlitz's result through computational analysis

Abstract

This paper contains two parts. The first is the solution of a challenge question, proposed by Etienne Ghys, on the determination of all maps from rational numbers to themselves such that the difference quotient (f(x)-f(y))/(x-y) is always a square. The second is the computer determination, done with the help of Stephane Gaubert, of a function of primes which plays a key role in the first part as a generalization of a result of Carlitz.