Repdigits base b as products of two Lucas numbers

Lucas Numbers Which Are Concatenations of Two Repdigits

“…This contradicts the assumption n > 1000. Finally, we search for the solutions to (1) in the range n ≤ 1000 by applying a program written in Mathematica and we obtain the solutions (n, L n ) ∈ {(6, 18), (7,29), (8,47), (9,76), (11,199), (12,322)}. We complete the proof.…”

“…In 2011, F. Luca and R. Oyono [2] concluded that there is no solution (m, n, s) to the Diophantine equation F s m + F s m+1 = F n for integers m ≥ 2, n ≥ 1, s ≥ 3 by applying linear form in logarithms. There are many papers in the literature which solve Diophantine equations related to Fibonacci numbers and Lucas numbers [3][4][5][6][7][8][9][10][11][12][13][14]. In 2013, D. Marques and A. Togbé [3] found all solutions (n, a, b, c) to the Diophantine equation F n = 2 a + 3 b + 5 c and L n = 2 a + 3 b + 5 c for integers n, a, b, c with 0 ≤ max{a, b} ≤ c. In 2019, B. D. Bitim [4] investigated the solutions (n, m, a) to the Diophantine equation L n − L m = 2 • 3 a for nonnegative integers n, m, a with n > m. Let p be a prime number and max{a, b} ≥ 2, in 2009, F. Luca and P. Stǎnicǎ [5] concluded that there are only finitely many positive integer solutions (n, p, a, b) to the Diophantine equation…”

“…, 9}, c = d, c > 0, s ≥ 1 and t ≥ 1. There are many papers in the literature on investigating Diophantine equations related to repdigits [8,9,[11][12][13][14][15][16][17][18][19][20][21]. In 2000, Luca [15] proved that if F n = a 10 m −1 two repdigits in balancing numbers, Padovan numbers and Tribonacci numbers, please refer to the literature [19][20][21] respectively.…”

Repdigits as Product of Fibonacci and Tribonacci Numbers

“…and some special types of linear recurrences (usually their product, sums, etc. ), where we refer the reader to [10][11][12][13][14][15][16][17][18] and references therein.…”

On Repdigits as Sums of Fibonacci and Tribonacci Numbers

“…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). For more about this subject, we refer the reader to [14][15][16][17][18][19][20][21][22][23][24] and references therein.…”