Repdigits Base $b$ as Difference of Two Fibonacci Numbers

Abstract

In this paper, we find all repdigits expressible as difference of two Fibonacci numbers in base $b$ for $2\leq b\leq10.$ The largest repdigits in base $b$, which can be written as difference of two Fibonacci numbers are \begin{align*}&F_{9}-F_{4}=34-3=31=(11111)_{2},~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(111111)_{3},\\&F_{14}-F_{7}=377-13=364=(222)_{4},~~ \text{ }F_{9}-F_{4}=34-3=31=(111)_{5},\\&F_{11}-F_{4}=89-3=86=(222)_{6},~~~~~~~~\text{ }F_{13}-F_{5}=233-5=228=(444)_{7},\\&F_{10}-F_{2}=55-1=54=(66)_{8},~~~~~~~~~~\text{ }F_{14}-F_{7}=377-13=364=(444)_{9},\end{align*} and $$F_{15}-F_{10}=610-55=555=(555)_{10}.$$

As a result, it is shown that the largest Fibonacci number which can be written as a sum of a repdigit and a Fibonacci number is $F_{15}=610=555+55=555+F_{10}.$