Type: Article

Common values of the arithmetic functions phi and sigma

Kevin Ford; Florian Luca; Carl Pomerance;

Abstract:

We show that the equation phi(a) = sigma(b) has infinitely many solutions, where phi is Euler's totient function and sigma is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erdos. Moreover, we show that, for some c > 0, there are infinitely many integers n such that phi(a) = n and sigma(b) = n, each having more than n(c) solutions. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of phi at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes