Solutions of f (n) = f (n + k) and s (n) = s (n + k)

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We show that for some even k 3570 and all k with 442720643463713815200|k, the equation f(n) = f(n+k) has infinitely many solutions n, where f is Euler's totient function. We also show that for a positive proportion of all k, the equation s(n) = s(n + k) has infinitelymany solutions n. The proofs rely on recent progress on the prime k-tuples conjecture by Zhang, Maynard, Tao, and PolyMath.

Original languageEnglish (US)
Pages (from-to)3561-3570
Number of pages10
JournalInternational Mathematics Research Notices
Issue number5
StatePublished - Mar 1 2022

ASJC Scopus subject areas

  • General Mathematics


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