Common divisors of totients of polynomial sequences

2019/09/24 by Brüdern, J., Soundararajan, K.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1909.10808

Abstract

Motivated by a question of Venkataramana, we consider the greatest common divisor of ϕ(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 n²+1 is not divisible by any prime ≡ 1 \bmod 2^m provided m is a large fixed integer.

Related

• On the greatest common divisor of integer parts of polynomials
• Greatest common divisors for polynomials in almost units and applications to linear recurrence sequences
• Prime and coprime values of polynomials
• Variations on a question concerning the degrees of divisors of xⁿ-1
• On the Common Prime Divisors of Polynomials