Orthogonal polynomials

Combinatorics

Discrete mathematics

Block (permutation group theory)

Mathematics

Logarithm

Fibonacci polynomials

Diophantine approximation

Mathematical analysis

Lucas sequence

Fibonacci number

Pisano period

Diophantine equation

Lucas number

Decimal

Difference polynomials

Algebraic number

Arithmetic

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.

Fibonacci Numbers with a Prescribed Block of Digits