Keith number

For information about Keith Olbermann's use of the term "Keith number", see Countdown with Keith Olbermann#The "Keith number".

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

$N=\sum_{i=0}^{n-1} 10^i {d_i},$

a sequence $S_N$ is formed with initial terms $d_{n-1}, d_{n-2},\ldots, d_1, d_0$ and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence $S_N$, then N is said to be a Keith number.

For example, taking 197 in such a way creates the sequence $1, 9, 7, 17, 33, 57, 107, 197, \ldots$. The first few Keith numbers are:

14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, 31331, 34285, 34348, 55604, 62662, 86935, 93993, 120284, 129106, 147640, 156146, 174680, 183186, 298320, 355419, 694280, 925993, 1084051, 7913837, 11436171 (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.

External links

• Keith Number - From MathWorld
• Keith Number - From Wolfram Mathworld