Diophantine equations involving the Euler totient function

  title={Diophantine equations involving the Euler totient function},
  author={J. C. Saunders},
  journal={arXiv: Number Theory},
  • J. C. Saunders
  • Published 5 February 2019
  • Mathematics
  • arXiv: Number Theory
We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences. 
On Fixed Points of Iterations Between the Order of Appearance and the Euler Totient Function
Let Fn be the nth Fibonacci number. The order of appearance z(n) of a natural number n is defined as the smallest positive integer k such that Fk≡0(modn). In this paper, we shall find all positiveExpand
Diophantine equations involving Euler function.
In this paper, we show that the equation $\varphi(|x^{m}-y^{m}|)=|x^{n}-y^{n}|$ has no nontrivial solutions in integers $x,y,m,n$ with $xy\neq0, m>0, n>0$ except for the solutionsExpand
The Euler Totient Function on Lucas Sequences
In 2009, Luca and Nicolae [14] proved that the only Fibonacci numbers whose Euler totient function is another Fibonacci number are 1, 2, and 3. In 2015, Faye and Luca [7] proved that the only PellExpand


The Euler Function of Fibonacci and Lucas Numbers and Factorials
Abstract : Here, we look at the Fibonacci and Lucas numbers whose Euler function is a factorial, as well as Lucas numbers whose Euler function is a product of power of two and power of three.
Existence of Primitive Divisors of Lucas and Lehmer Numbers
We prove that for n > 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.
Perfect fibonacci and lucas numbers
In this note, we show that the classical Fibonacci and Lucas sequence do not contain any perfect number.
Square values of Euler's function
We show that almost all squares are missing from the range of Euler’s φfunction.
Unsolved Problems in Number Theory
This monograph contains discussions of hundreds of open questions, organized into 185 different topics. They represent aspects of number theory and are organized into six categories: prime numbers,Expand
Common values of the arithmetic functions ϕ and σ
We show that the equation φ(a )= σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sum-of-divisors function. This proves a fifty-year-old conjecture of Erd˝os.Expand
On polynomial-factorial diophantine equations
We study equations of the form P(x) = n! and show that for some classes of polynomials P the equation has only finitely many solutions. This is the case, say, if P is irreducible (of degree greaterExpand
Here, we show that if u0 = 0, u1 = 1, and un+2 = run+1 + sun for all n ≥ 0 is the Lucas sequence with s ∈ {±1}, then there are only finitely many effectively computable n such that φ(|un|) is a powerExpand
Powers of Two in Generalized Fibonacci Sequences
The k generalized Fibonacci sequence F (k) n n resembles the Fi- bonacci sequence in that it starts with 0;:::; 0; 1 (k terms) and each term af- terwards is the sum of the k preceding terms. In thisExpand
Rendiconti del circolo matematico di Palermo
1. After du Bois-Reymond [7] had constructed the example of a continuous function whose Fourier series diverges at apoint Jordan [12] gave a new criterion for convergence of Fourier series whichExpand