On the x-coordinates of Pell equations that are sums of two Padovan numbers

EasyChair Preprint no. 2635

Abstract

Let $ (P_{n})_{n\geq 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1, ~P_2=1$, and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\geq 0 $. In this paper, we find all positive square-free integers, $ d $ such that the Pell equations $ x^2-dy^2 = \pm 1 $, $ X^2-dY^2=\pm 4 $, have at least two positive integer solutions $ (x,y) $ and $(x^{\prime}, y^{\prime})$, $ (X,Y) $ and $(X^{\prime}, Y^{\prime})$, respectively, such that each of $ x, ~x^{\prime}, ~X$, and $ X^{\prime} $ is a sum of two Padovan numbers.

Keyphrases: linear forms in logarithms, Padovan numbers, Pell equations, reduction method

@Booklet{EasyChair:2635,
  author = {Mahadi Ddamulira},
  title = {On the x-coordinates of Pell equations that are sums of two Padovan numbers},
  howpublished = {EasyChair Preprint no. 2635},

  year = {EasyChair, 2020}}