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 OEIS|id=A007629

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

* [http://mathworld.wolfram.com/KeithNumber.html Keith Number - From MathWorld]