In mathematics, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is an integer N>9 that appears as a term in a linear recurrence relation with initial terms based on its own digits. Given an n-digit number <math>N=\sum_{i=0}^{n-1} 10^i {d_i},</math> a sequence <math>S_N</math> is formed with initial terms <math>d_{n-1}, d_{n-2},\ldots, d_1, d_0</math> and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence <math>S_N</math>, then N is said to be a Keith number. For example, taking 197 in such a way creates the sequence <math>1, 9, 7, 17, 33, 57, 107, 197, \ldots</math>. The first few Keith numbers are: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909 (sequence A007629 in OEIS) Whether or not there are infinitely many Keith numbers is currently a matter of speculation. There are only 71 Keith numbers below 1019, making them much rarer than prime numbers. Mike Keith is a mathematician who published a paper on these numbers titled "Repfigit Numbers" in a 1987 issue of the Journal of Recreational Mathematics.
