Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k). (26th August 2020) - Ford, Kevin
Abstract: We show that for some even $k\leqslant 3570$ and all $k$ with $442720643463713815200|k$, the equation $φ (n)=φ (n+k)$ has infinitely many solutions $n$, where $φ $ is Euler's totient function. We also show that for a positive proportion of all $k$, the equation $σ (n)=σ (n+k)$ has infinitely many solutions $n$ . The proofs rely on recent progress on the prime $k$ -tuples conjecture by Zhang, Maynard, Tao, and PolyMath.