Common divisors of totients of polynomial sequences

TL;DR

This paper investigates the common divisors of Euler's totient function applied to polynomial sequences, establishing bounds under certain hypotheses and for specific polynomial classes, with implications for prime divisibility properties.

Contribution

It provides bounds on the gcd of totients of polynomial values under Schinzel's hypothesis and unconditionally for quadratic and linear-splitting polynomials.

Findings

Bounded gcd of totients for polynomials assuming Schinzel's hypothesis.

Unconditional bounds for quadratic and linear-splitting polynomials.

Existence of infinitely many integers not divisible by certain primes in specific residue classes.

Abstract

Motivated by a question of Venkataramana, we consider the greatest common divisor of $\phi(f(n))$ where $f$ is a primitive polynomial with integer coefficients, and $n$ ranges over all natural numbers. Assuming Schinzel's hypothesis, we establish that this gcd may be bounded just in terms of the degree of the polynomial $f$. Unconditionally we establish such a bound for quadratic polynomials, as well as polynomials that split completely into linear factors. The paper also addresses a question of Calegari, and establishes that there are infinitely many $n$ such that $n^2+1$ is not divisible by any prime $\equiv 1 \bmod 2^m$ provided $m$ is a large fixed integer.