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   null (Ed.)

   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. 

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   https://doi.org/10.1007/JHEP06(2021)137  

   Mizuk, R. ; Bondar, A. ; Adachi, I. ; Aihara, H. ; Al Said, S. ; Asner, D. M. ; Atmacan, H. ; Aulchenko, V. ; Aushev, T. ; Ayad, R. ; et al ( June 2021 , Journal of High Energy Physics)

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