Solutions of phi (n) = phi (n plus k) and sigma (n) = sigma (n plus k)

We show that for some even k <= 3570 and all k with 442720643463713815200|k, the equation phi(n) = phi(n+ k) has infinitely many solutions n, where phi is Euler's totient function. We also show that for a positive proportion of all k, the equation sigma(n) = sigma(n + k) has infinitelymany solutions n. The proofs rely on recent progress on the prime k-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.