On Diophantine equations involving Lucas sequences

Abstract

In this paper, we shall study the Diophantine equation un = R(m)P(m)Q(m), where un is a Lucas sequence and R, P and Q are polynomials (under weak assumptions).

On Repdigits as Sums of Fibonacci and Tribonacci Numbers

“…c 3 c 2 c 1 (thus it can be said that they are "symmetrical" with respect to an axis of symmetry). The first 19th palindromic numbers are 0, 1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88,99 and clearly they are a repdigits type. A number n is called repdigit if it has only one repeated digit in its decimal expansion.…”

“…(it is an easy exercise that if R is prime, then so is ). There are many articles that address Diophantine equations concerning Fibonacci and Lucas numbers (see, e.g., [2][3][4][5][6][7][8][9][10][11][12][13]). In the last years, many authors have worked on Diophantine problems related to repdigits (e.g., their sums, concatenations) and linear recurrences (e.g., their product, sums).…”

Repdigits as Product of Fibonacci and Tribonacci Numbers

“…It is also important to notice that their studies contributed strongly to the advance of mathematics. There are many articles that address Diophantine equations concerning Fibonacci and Lucas numbers (see, e.g., [1][2][3][4][5][6][7][8]). The linear forms in logarithms, which were probably firstly used for solving Diophantine equations in Dujella and Jadrijević [9], have proved to be a very effective tool for finding solutions to all these equations.…”