The Carlitz group of the rationals
Alain Connes

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.
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Taxonomy
TopicsMathematics and Applications · History and Theory of Mathematics
