Common values of the arithmetic functions φ and σ

We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler's totient function and σ is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erds. Moreover, we show that, for some c > 0, there are infinitely many integers n such that φ(a) = n and σ(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 φ at p, and on a result of Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes.