  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.
