Editing Keith number (section)

== Definition ==
Let <math>n</math> be a natural number, let <math>k = \lfloor \log_{b}{n} \rfloor + 1</math> be the number of digits of <math>n</math> in base <math>b</math>, and let 
:<math>d_i = \frac{n \bmod b^{i + 1} - n \bmod b^{i}}{b^{i}}</math>
be the value of each digit of <math>n</math>.

We define the sequence <math>S(i)</math> by a [[linear recurrence relation]]. For <math>0 \leq i < k</math>, 
:<math>S(i) = d_{k - i - 1}</math>
and for <math>i \geq k</math>
:<math>S(i) = \sum_{j = 0}^{k} S(i - k + j)</math>

If there exists an <math>i</math> such that <math>S(i) = n</math>, then <math>n</math> is said to be a '''Keith number'''. 

For example, 88 is a Keith number in [[base 6]], as 
:<math>S(0) = d_{3 - 0 - 1} = d_2 = \frac{88 \bmod 6^{2 + 1} - 88 \bmod 6^{2}}{6^{2}} = \frac{88 \bmod 216 - 88 \bmod 36}{36} = \frac{88 - 16}{36} = \frac{72}{36} = 2</math>
:<math>S(1) = d_{3 - 1 - 1} = d_1 = \frac{88 \bmod 6^{1 + 1} - 88 \bmod 6^{1}}{6^{1}} = \frac{88 \bmod 36 - 88 \bmod 6}{6} = \frac{16 - 4}{6} = \frac{12}{6} = 2</math>
:<math>S(2) = d_{3 - 2 - 1} = d_0 = \frac{88 \bmod 6^{0 + 1} - 88 \bmod 6^{0}}{6^{0}} = \frac{88 \bmod 6 - 88 \bmod 1}{1} = \frac{4 - 0}{1} = \frac{4}{1} = 4</math>
and the entire sequence
:<math>S(i) = \{2, 2, 4, 8, 14, 26, 48, 88, 162, \ldots\}</math>
and <math>S(7) = 88</math>.

===Finding Keith numbers===
Whether or not there are infinitely many Keith numbers in a particular base <math>b</math> is currently a matter of speculation. Keith numbers are rare and hard to find. They can be found by exhaustive search, and no more efficient algorithm is known.<ref>{{cite web | last1 = Earls | first1 = Jason | last2 = Lichtblau | first2 = Daniel | last3 = Weisstein | first3 = Eric W. | author-link = Eric W. Weisstein| title = Keith Number | publisher = [[MathWorld]] | url = http://mathworld.wolfram.com/KeithNumber.html }}</ref>
According to Keith, in [[base 10]], on average <math>\textstyle\frac{9}{10}\log_2{10}\approx 2.99</math> Keith numbers are expected between successive [[power of 10|powers of 10]].<ref name="keith_web">{{cite web | author-link = Mike Keith (mathematician) | first = Mike | last = Keith | title = Keith Numbers | url = http://www.cadaeic.net/keithnum.htm }}</ref> Known results seem to support this.