Fibonacci Numbers with a Prescribed Block of Digits

Pavel Trojovský

doi:10.3390/math8040639

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.

Highlights

Let ( Fn )n≥0 be the Fibonacci sequence given by second-order recurrence Fn+2 = Fn+1 + Fn, for n ≥ 0, with initial conditions F0 = 0 and F1 = 1
Luca [3] who showed that the largest Fibonacci number with only one distinct digit is F10 = 55
Many authors worked on repdigits as expressions related to sum, product of terms of binary recurrences

Summary

Introduction

Luca [3] who showed that the largest Fibonacci number with only one distinct digit is F10 = 55. 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). The related problem of finding all Fibonacci numbers with only two distinct digits remains open. Our main result searches for all Fibonacci numbers of the form ab . Our proof combines two deep techniques in number theory, namely, the Baker’s theory on linear forms in logarithms and some tools from Diophantine approximation

Auxiliary Results

Finding a Bound on N

Reducing the Bound

Conclusions