Keith number
Encyclopedia
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
a sequence
is formed with initial terms 
and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence 
, then N is said to be a Keith number.
For example, taking 197 in such a way creates the sequence
. 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
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 number
s.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Keith number or repfigit number (short for repetitive Fibonacci-like digit) is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
N > 9 that appears as a term in a linear recurrence relation
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence, once one or more initial terms are given: each further term of the sequence is defined as a function of the preceding terms....
with initial terms based on its own digits. Given an n-digit number
a sequence
is formed with initial terms and with a general term produced as the sum of the previous n terms. If the number N appears in the sequence , then N is said to be a Keith number.For example, taking 197 in such a way creates the sequence
. The first few Keith numbers are:14
14 (number)
14 is the natural number following 13 and preceding 15.In speech, the numbers 14 and 40 are often confused. When carefully enunciated, they differ in which syllable is stressed: 14 vs 40...
, 19
19 (number)
19 is the natural number following 18 and preceding 20. It is a prime number.In English speech, the numbers 19 and 90 are often confused. When carefully enunciated, they differ in which syllable is stressed: 19 vs 90...
, 28
28 (number)
28 is the natural number following 27 and preceding 29.-In mathematics:It is a composite number, its proper divisors being 1, 2, 4, 7, and 14....
, 47
47 (number)
47 is the natural number following 46 and preceding 48.-In mathematics:Forty-seven is the fifteenth prime number, a safe prime, the thirteenth supersingular prime, and the sixth Lucas prime. Forty-seven is a highly cototient number...
, 61
61 (number)
61 is the natural number following 60 and preceding 62.-In mathematics:It is the 18th prime number. The previous is 59, with which it comprises a twin prime. Sixty-one is a cuban prime of the form p = / , x = y + 1....
, 75
75 (number)
75 is the natural number following 74 and preceding 76.-In mathematics:As the sum of the first five pentagonal numbers, seventy-five is a pentagonal pyramidal number. It is also an enneagonal number....
, 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
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 number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s.
Mike Keith
Mike Keith (mathematician)
Mike Keith is an American mathematician, software engineer, and author of works of constrained writing....
is a mathematician who published a paper on these numbers titled "Repfigit Numbers" in a 1987 issue of the Journal of Recreational Mathematics.