Journal of Integer Sequences, Vol. 26 (2023), Article 23.1.6

On the Minimal Number of Solutions of the Equation φ(n+k) = Mφ(n), M = 1,2

Matteo Ferrari
Dipartimento di Scienze Matematiche "G.L. Lagrange"
Politecnico di Torino
Corso Duca degli Abruzzi 24
10138 Torino

Lorenzo Sillari
Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Via Bonomea 265
34136 Trieste


We fix a positive integer k and look for solutions nN of the equations φ(n + k) = φ(n) and φ(n + k) = 2φ(n). For k ≤ 12 · 10100, we prove that Fermat primes can be used to build five solutions for the first equation when k is even, and five for the second one when k is odd. Furthermore, for k ≤ 4 · 1058, we show that for the second equation there are at least three solutions when k is even. Our work increases the previously known minimal number of solutions for both equations.

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(Concerned with sequences A001259 A358717 A358718 A358719.)

Received September 7 2022; revised versions received December 27 2022; January 14 2023. Published in Journal of Integer Sequences, January 15 2023.

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