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 k3570 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.