In recreational mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is a natural number ${\displaystyle n}$ in a given number base ${\displaystyle b}$ with ${\displaystyle k}$ digits such that when a sequence is created such that the first ${\displaystyle k}$ terms are the ${\displaystyle k}$ digits of ${\displaystyle n}$ and each subsequent term is the sum of the previous ${\displaystyle k}$ terms, ${\displaystyle n}$ is part of the sequence. Keith numbers were introduced by Mike Keith in 1987. They are computationally very challenging to find, with only about 125 known.

Let ${\displaystyle n}$ be a natural number, let ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$ be the number of digits of ${\displaystyle n}$ in base ${\displaystyle b}$, and let

be the value of each digit of ${\displaystyle n}$.

We define the sequence ${\displaystyle S(i)}$ by a linear recurrence relation. For ${\displaystyle 0\leq i<k}$,

and for ${\displaystyle i\geq k}$

If there exists an ${\displaystyle i}$ such that ${\displaystyle S(i)=n}$, then ${\displaystyle n}$ is said to be a Keith number.

For example, 88 is a Keith number in base 6, as

and the entire sequence