Positive
Solutions of ϕ (n) = ϕ (n + k) and σ (n) = σ (n + k)
Abstract We show that for some even $k\leqslant 3570$ and all $k$ with $442720643463713815200k$, 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.
 Award ID(s):
 1802139
 Publication Date:
 NSFPAR ID:
 10252047
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Sponsoring Org:
 National Science Foundation
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