Definition:Keith Number

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Definition

Let $K \in \Z_{>9}$ be an $n$-digit integer, where $n > 1$.

Let $A = \left({a_1, a_2, \ldots, a_n}\right)$ be the digits of $K$ in order.

Let $F_A$ be the Fibonacci-like sequence on $A$.


Then $K$ is a Keith number if and only if $K$ occurs somewhere in $F_A$.


Sequence

The sequence of Keith numbers begins:

$14, 19, 28, 47, 61, 75, 197, 742, 1104, \ldots$


Examples

Keith Number: $14$

$14$ is a Keith number:

$1, 4, 5, 9, 14, \ldots$


Keith Number: $19$

$19$ is a Keith number:

$1, 9, 10, 19, \ldots$


Keith Number: $197$

$197$ is a Keith number:

$1, 9, 7, 17, 33, 57, 107, 197, \ldots$


Keith Number: $129 \, 572 \, 008$

$129 \, 572 \, 008$ is a Keith number:

$1, 2, 9, 5, 7, 2, 0, 0, 8, \ldots, 32 \, 456 \, 930, 64 \, 849 \, 899, 129 \, 572 \, 008, \ldots$


Keith Number: $251 \, 133 \, 297$

$251 \, 133 \, 297$ is a Keith number:

$2, 5, 1, 1, 3, 3, 2, 9, 7, 33, 64, 123, 245, 489, 975, 1947, 3892, 7775,$
$15 \, 543, 31 \, 053, 62 \, 042, 123 \, 961, 247 \, 677, 494 \, 865, 988 \, 755,$
$1 \, 975 \, 563, 3 \, 947 \, 234, 7 \, 886 \, 693, 15 \, 757 \, 843, 31 \, 484 \, 633,$
$62 \, 907 \, 224, 125 \, 690 \, 487, 251 \, 133 \, 297, \ldots$


Keith Number: $754 \, 788 \, 753 \, 590 \, 897$

$754 \, 788 \, 753 \, 590 \, 897$ is a Keith number:

$7, 5, 4, 7, 8, 8, 7, 5, 3, 5, 9, 0, 8, 9, 7, \ldots, $


Also known as

A Keith number is referred to by some sources as a repfigit (repetitive fibonacci-like digit) number.


Source of Name

This entry was named for Michael Keith.


Historical Note

Keith numbers were first introduced by Mike Keith in 1987: Repfigit Numbers (J. Recr. Math. Vol. 19: 41 – 42).

The author coined the name repfigit, but they have since been assigned the name of their creator.


They proved a popular topic, and in $1994$, in Volume $26$, Number $3$ of the Journal of Recreational Mathematics, several articles on them appeared.

The number of known Keith numbers is gradually extended, and as of time of writing ($5$th February $2017$) all Keith numbers have been identified up to $10^{29}$.


Mike Keith continues to maintain his webpage on the subject, which can be found at http://www.cadaeic.net/keithnum.htm.


Sources

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