Carmichael numbers with a totient of the form a 2 + nb 2

Abstract

Let φ be the Euler function. Fix \({\ell\in{\mathbb N}}\) , and let \({{\fancyscript P}}\) be an arbitrary set of primes of positive lower natural density. Using a variant of the Alford–Granville–Pomerance construction, we show that there are infinitely many Carmichael numbers N with a totient of the form \({\varphi(N)=m^\ell \tilde m}\) , where \({m,\tilde m\in{\mathbb N}}\) and \({\tilde m}\) is a nonempty product of primes from the set \({{\fancyscript P}}\) . In particular, for any fixed natural number n, there are infinitely many Carmichael numbers N such that φ(N) = a2 + nb2 for some positive integers a and b.