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We show that for some even   $k\leqslant 3570$   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 infinitely many solutions   $n$  . The proofs rely on recent progress on the prime   $k$  -tuples conjecture by Zhang, Maynard, Tao, and PolyMath.

Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)