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

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Matteo Ferrari

Dipartimento di Scienze Matematiche "G.L. Lagrange"

Politecnico di Torino

Corso Duca degli Abruzzi 24

10138 Torino

Italy

Lorenzo Sillari

Scuola Internazionale Superiore di Studi Avanzati (SISSA)

Via Bonomea 265

34136 Trieste

Italy

**Abstract:**

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 · 10^{100},
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 · 10^{58},
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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