Common divisors of totients of polynomial sequences

This paper considers two related problems connected to sieving values of polynomials by primes lying in certain arithmetic progressions. The first problem was raised by Calegari [ 1 ], who asked in a blog post whether one can show that there are infinitely many n such that \(n^2+1\) is not divisible by any prime \(p\equiv 1\,\mathrm{mod~}2^m\) where m is some fixed large integer. One expects that the polynomial \((4n+2)^2 + 1= 16n^2+16 n+5\) takes prime values infinitely often, so that there should be infinitely many values of n with \(n^2+1\) divisible by no prime \(\equiv 1\, \mathrm{mod~}8\) . In Theorem 4 we shall give a resolution of Calegari’s question for irreducible quadratic polynomials.