  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 50-year old conjecture of Erdos. Moreover, we show that
there are infinitely many integers n such that phi(a)=n and sigma(b)=n each
have more than n^c solutions, for some c>0. 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.
