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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 Erdős. 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.

Common values of the arithmetic functions ϕ and σ