We show that there are infinitely many integers which are both values
of Euler's function
and of the sum of divisors function, settling a 50-year old conjecture of
Erdös. The main
new ingredient is a bound for counts of prime chains in a recent paper of
the speaker, S.
Konyagin and F. Luca.
This is joint work with Florian Luca and Carl Pomerance.