Solutions ofϕ(n) =ϕ(n + k) andσ(n) =σ(n + k)

DOI10.1093/IMRN/RNAA218zbMATH Open1494.11005arXiv2002.12155OpenAlexW3081881205MaRDI QIDQ5862788

Kevin Ford

Publication date: 10 March 2022

Published in: IMRN. International Mathematics Research Notices (Search for Journal in Brave)

Abstract: We show that for some $kle3570$ 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.

Full work available at URL: https://arxiv.org/abs/2002.12155

zbMATH Keywords

divisor functionEuler functionprime \(k\)-tuples conjecturegeneralized prime twinsPolyMath

Mathematics Subject Classification ID

Arithmetic functions; related numbers; inversion formulas (11A25) Difference equations (39A99)

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