A local-global principle for power maps

TL;DR

This paper investigates the conditions under which a function behaving like a power map locally at many primes must be a global power map, providing progress towards a longstanding conjecture in number theory.

Contribution

The authors prove that if a function acts as a local power map at a set of primes with positive density, then it is necessarily a global power map, advancing the understanding of the local-global principle.

Findings

If f is a local power map at infinitely many primes, then f is a global power map.

The main theorem shows that positive density of primes where f is a local power map implies f is global.

Progress towards the conjecture of Fabrykowski and Subbarao on power maps.

Abstract

Let f be a function from the set of rational numbers into itself. We call f a global power map if f(n) = n^k for some integer exponent k. We call f a local power map at the prime number p if f induces a well-defined group homomorphism on the multiplicative group of integers modulo p. We conjecture that if f is a local power map at an infinite number of primes p, then f must be a global power map. Our main theorem implies that if f is a local power map at every prime p in a set with positive upper density relative to the set of all primes, then f must be a global power map. In particular, this represents progress towards a conjecture of Fabrykowski and Subbarao.