The Carlitz group of the rationals

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.