Keith Number

A Keith number is an $n$-digit Integer $N$ such that if a Fibonacci-like sequence (in which each term in the sequence is the sum of the $n$ previous terms) is formed with the first $n$ terms taken as the decimal digits of the number $N$, then $N$ itself occurs as a term in the sequence. For example, 197 is a Keith number since it generates the sequence 1, 9, 7, 17, 33, 57, 107, 197, ... (Keith). Keith numbers are also called Repfigit Numbers.

There is no known general technique for finding Keith numbers except by exhaustive search. Keith numbers are much rarer than the Primes, with only 52 Keith numbers with $<15$ digits: 14, 19, 28, 47, 61, 75, 197, 742, 1104, 1537, 2208, 2580, 3684, 4788, 7385, 7647, 7909, ... (Sloane's A007629). In addition, three 15-digit Keith numbers are known (Keith 1994). It is not known if there are an Infinite number of Keith numbers.

References

Esche, H. A. ``Non-Decimal Replicating Fibonacci Digits.'' J. Recr. Math. 26, 193-194, 1994.

Heleen, B. ``Finding Repfigits--A New Approach.'' J. Recr. Math. 26, 184-187, 1994.

Keith, M. ``Repfigit Numbers.'' J. Recr. Math. 19, 41-42, 1987.

Keith, M. ``All Repfigit Numbers Less than 100 Billion ($10^{11}$).'' J. Recr. Math. 26, 181-184, 1994.

Keith, M. ``Keith Numbers.'' http://users.aol.com/s6sj7gt/mikekeit.htm.

Robinson, N. M. ``All Known Replicating Fibonacci Digits Less than One Thousand Billion ($10^{12}$).'' J. Recr. Math. 26, 188-191, 1994.

Shirriff, K. ``Computing Replicating Fibonacci Digits.'' J. Recr. Math. 26, 191-193, 1994.

Sloane, N. J. A. Sequence A007629 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

``Table: Repfigit Numbers (Base $10^*$) Less than $10^{15}$.'' J. Recr. Math. 26, 195, 1994.