Fibonacci Numbers with a Prescribed Block of Digits

Abstract

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

Padovan numbers which are palindromic concatenations of two distinct repdigits

“…[1], the second author considered all Padovan numbers that can be written as a concatenation of two distinct repdigits and showed that the largest such number is P 21 = 200. More specifically, it was shown that if P n is a solution of the Diophantine equation Other related interesting results in this research direction include: the result of Bednařík and Trojovská [3], the result of Boussayoud, et al [4], the result of Bravo and Luca [5], the result of the second author [7], the result of Erduvan and Keskin [11], the result of Rayaguru and Panda [16], the results of Trojovský [17,18], and the result of Qu and Zeng [15]. A natural continuation of the result in Ref.…”

Tribonacci numbers that are concatenations of two repdigits

“…This paper is inspired by the results of Alahmadi et al [ 1 ], in which they show that the only Fibonacci numbers that are concatenations of two repdigits are , and Rayaguru and Panda [ 10 ], who showed that is the only balancing number that can be written as a concatenation of two repdigits. Other related interesting results in this direction include: the result of Bravo and Luca [ 3 ], the result of Trojovský [ 11 ], the result of Qu and Zeng [ 9 ], and the result of Boussayoud et al [ 2 ].…”

Lucas Numbers Which Are Concatenations of Two Repdigits

“…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…”