Fibonacci numbers which are concatenations of two repdigits

Abstract

We show that the only Fibonacci numbers that are concatenations of two repdigits are 13,

Products of $k$-Fibonacci numbers which are rep-digits

“…. , α (k) for all roots of Ψ k (x) with the convention that α (1) := α. We now consider, for an integer k ≥ 2, the function…”

“…Since then this result was generalised and extended in various directions. For example, in [1], we found all Fibonacci numbers which are concatenations of two rep-digits. In [14],…”

Padovan numbers which are palindromic concatenations of two distinct repdigits

“…In [1], the authors considered Fibonnaci numbers which are concatenations of two repdigits (in base 10) and showed that the largest such number is F 14 = 377. Recently, diophantine equations involving Padovan numbers and repdigits have also been studied.…”

Fibonacci Numbers with a Prescribed Block of Digits

“…A first result in this direction was proved, in 2000, by F. Luca [3] who showed that the largest Fibonacci number with only one distinct digit is F 10 = 55. After this, many authors worked on repdigits (i.e., integers having only one distinct digit in its decimal expansion) as expressions related to sum, product of terms of binary recurrences (see [4][5][6][7][8][9][10][11][12][13] and references therein). However, the related problem of finding all Fibonacci numbers with only two distinct digits remains open.…”