On the Minimal Number of Solutions of the
Equation φ(n+k) = Mφ(n), M = 1,2
Dipartimento di Scienze Matematiche "G.L. Lagrange"
Politecnico di Torino
Corso Duca degli Abruzzi 24
Scuola Internazionale Superiore di Studi Avanzati (SISSA)
Via Bonomea 265
We fix a positive integer k and look for solutions
n ∈ N 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.
Full version: pdf,
(Concerned with sequences
Received September 7 2022; revised versions received December 27 2022; January 14 2023.
Published in Journal of Integer Sequences,
January 15 2023.
Journal of Integer Sequences home page