On the Euler Function of Y-Coordinates of Pell Equations and Repdigits

2022 ◽  
Vol 77 (2) ◽  
Author(s):  
Adel Alahmadi ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Pell Equations

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

Cubic Pell Equations

2018 ◽  
pp. 205-245
Author(s):  
Samuel A. Hambleton ◽  
Hugh C. Williams
Keyword(s):  
Pell Equations

scholarly journals On the number of solutions of simultaneous Pell equations

2002 ◽  
Vol 101 (3) ◽  
pp. 215-221 ◽  
Author(s):  
Pingzhi Yuan
Keyword(s):  
Number Of Solutions ◽  
Pell Equations ◽  
Simultaneous Pell Equations

scholarly journals Pseudoprime values of the Fibonacci sequence, polynomials and the Euler function

2006 ◽  
Vol 17 (4) ◽  
pp. 611-625 ◽  
Author(s):  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Fibonacci Sequence ◽  
Euler Function

On the sum of the first n values of the Euler function

2014 ◽  
Vol 163 (3) ◽  
pp. 199-201 ◽  
Author(s):  
R. Balasubramanian ◽  
Florian Luca ◽  
Dimbinaina Ralaivaosaona
Keyword(s):  
Euler Function

Carmichael Numbers with a Square Totient

2009 ◽  
Vol 52 (1) ◽  
pp. 3-8 ◽  
Author(s):  
W. D. Banks
Keyword(s):  
Euler Function ◽  
Carmichael Numbers

AbstractLet φ denote the Euler function. In this paper, we show that for all large x there are more than x0.33 Carmichael numbers n ⩽ x with the property that φ(n) is a perfect square. We also obtain similar results for higher powers.

Orthogonal-type polynomials and Pell equations

2013 ◽  
Vol 24 (10) ◽  
pp. 796-806 ◽  
Author(s):  
James C. Griffin ◽  
Gajath Gunatillake
Keyword(s):  
Pell Equations
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