Power-Free Values of Polynomials

TL;DR

This paper establishes an asymptotic formula for counting primes p less than x for which the polynomial f(p) is (d-1)-free, extending understanding of prime values of polynomials.

Contribution

It provides the first asymptotic estimate for the distribution of primes p where f(p) is (d-1)-free, for irreducible polynomials of degree at least 3 with no fixed prime divisor.

Findings

Asymptotic formula derived for primes p with f(p) (d-1)-free

Results apply to irreducible polynomials of degree ≥ 3

No fixed prime divisor condition essential for the result

Abstract

Let f be an irreducible polynomial of degree d>=3 with no fixed prime divisor. We derive an asymptotic formula for the number of primes p<x such that f(p) is (d-1)-free.