Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k). (26th August 2020)

• Record Type: Journal Article
• Authors: Ford, Kevin
• Is Part Of: International mathematics research notices. Volume 2022:Number 5(2022)
• Page Start: 3561
• Page End: 3570
• Publication Date: 2020-08-26
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnaa218
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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.