Zeckendorf representations with at most two terms to x-coordinates of Pell equations

Abstract

In this paper, we find all positive squarefree integers d such that the Pell equation X 2 − dY 2 = ±1 has at least two positive integer solutions (X, Y ) and (X ′ , Y ′ ) such that both X and X ′ have Zeckendorf representations with at most two terms.This paper has been accepted for publication in SCIENCE CHINA Mathematics.

The x-coordinates of Pell equations and sums of two Fibonacci numbers II

“…In this paper, we study a problem related to that of Gómez and Luca [2], but for the Pell equation (1) instead of x 2 − dy 2 = ±1. Before formulating our main theorem, let us notice that our problem is a bit different from the previous ones in that there are infinitely many d's such that the equation x k = F n + F m with n ≥ m ≥ 0 has at least two solutions (m, n, k).…”

“…We do a suitable cross product between Γ (1) 2 , Γ (2) 2 and k 1 , k 2 to eliminate the term involving log δ in the above linear forms in logarithms:…”

“…Several other related problems have been studied where x k belongs to some interesting positive integer sequences. For example, see [2,5,6,7,9,11,12,13,14,15].…”

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

“…Using the Eqs. (8), (12), (19), and (21), we have 1 2 ðd k þ r k Þ ¼ P n þ P m ¼ aa n þ eðnÞ þ aa m þ eðmÞ:…”

“…Several other related problems have been studied where x k belongs to some interesting positive integer sequences. For example, see [8,9,11,12,[14][15][16]18].…”