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Title: 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 $442720643463713815200|k$, 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.
Authors:
Award ID(s):
1802139
Publication Date:
NSF-PAR ID:
10252047
Journal Name:
International Mathematics Research Notices
ISSN:
1073-7928
Sponsoring Org:
National Science Foundation
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