Keith number

In entertainment mathematics , a Keith number (English Keith number , but also repfigit number (short for rep etitive Fi bonacci-like di git )) is a natural number that defines a special mathematical sequence through its digits and is contained in it. ${\displaystyle n\in \mathbb {N} }$

Be a natural number with digits , so ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k}$ ${\displaystyle d_{k-1},d_{k-2},\ldots ,d_{1},d_{0}}$

${\displaystyle n=\sum _{i=0}^{k-1}10^{i}{d_{i}}}$

Be a math sequence that starts with the values . Each further sequence element is the sum of the preceding sequence elements. If the number is in this sequence , then it is a Keith number . Because single-digit numbers trivially fulfill this property, they are usually not accepted as Keith numbers. So it has to be. ${\displaystyle S_{n}}$ ${\displaystyle d_{k-1},d_{k-2},\ldots ,d_{1},d_{0}}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle n\geq 10}$

The mathematician Mike Keith ( en ) was the first to deal with these numbers in 1997.

There are no known quick techniques for calculating Keith numbers other than the above method.

Examples

• Be the -digit number . Then the first members of the sequence are as follows: ${\displaystyle n}$ ${\displaystyle 3}$ ${\displaystyle n=742}$ ${\displaystyle s_{n}}$ ${\displaystyle S_{n}}$

7 , 4 , 2 , 13, 19, 34, 66, 119, 219, 404, 742 , 1365, 2511, 4618, 8494, 15623, 28735, 52852, ...

The following term is the sum of the three preceding terms and . So it is . For example is . Because the -digit number is included in this sequence, it is a Keith number. ${\displaystyle s_{i}}$ ${\displaystyle s_{i-3},s_{i-2}}$ ${\displaystyle s_{i-1}}$ ${\displaystyle s_{i}=s_{i-3}+s_{i-2}+s_{i-1}}$ ${\displaystyle 742=119+219+404}$ ${\displaystyle 3}$ ${\displaystyle n=742}$ ${\displaystyle n=742}$

• Be the -digit number . Then the first members of the sequence are as follows: ${\displaystyle n}$ ${\displaystyle 5}$ ${\displaystyle n=34285}$ ${\displaystyle s_{n}}$ ${\displaystyle S_{n}}$

3 , 4 , 2 , 8 , 5 , 22, 41, 78, 154, 300, 595, 1168, 2295, 4512, 8870, 17440, 34285 , 67402, 132509, 260506, 512142, 1006844, ...

The following term is the sum of the five preceding terms and . So it is . For example is . Because the -digit number is included in this sequence, it is a Keith number. ${\displaystyle s_{i}}$ ${\displaystyle s_{i-5},s_{i-4},s_{i-3},s_{i-2}}$ ${\displaystyle s_{i-1}}$ ${\displaystyle s_{i}=s_{i-5}+s_{i-4}+s_{i-3}+s_{i-2}+s_{i-1}}$ ${\displaystyle 4512=154+300+595+1168+2295}$ ${\displaystyle 5}$ ${\displaystyle n=34285}$ ${\displaystyle n=34285}$

• The first 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, 33445755, 44121607, ... (sequence A007629 in OEIS )

If you add the one-digit trivial Keith numbers, you get the sequence A130010 in OEIS .

• The number of Keith numbers with digits can be found in the following list (the zero at the beginning only applies if you do not include the one-digit trivial Keith numbers): ${\displaystyle k=1,2,3,\ldots }$

0, 6, 2, 9, 7, 10, 2, 3, 2, 0, 2, 4, 2, 3, 3, 3, 5 , 3, 5, 3, 1, 1, 3, 1, 1, 3, 7, 1, 2, 5, 2, 4, 6, 3, ... (sequence A050235 in OEIS )

Example:

The 17th position of the list above shows the number . That means, there are exactly Keith numbers which have 17 digits (for which the following applies). ${\displaystyle 5}$ ${\displaystyle 5}$ ${\displaystyle 10^{17}\leq n<10^{18}}$

• There are only 99 Keith numbers that have 30 digits or less. The 99th Keith number has 30 digits and is . ${\displaystyle n=534139807526361917710268232010}$
• The largest known Keith number currently (as of December 30, 2018) is the following:

${\displaystyle n=5752090994058710841670361653731519}$

This number has 34 digits and was discovered by Daniel Lichtblau on August 26, 2009. ${\displaystyle n}$

properties

• There are no Keith numbers that are repdigits at the same time (i.e. only consist of the same digits).

assumptions

• It is believed that there are infinitely many Keith numbers.

Keith claims based on experience that there are Keith numbers between and for . ${\displaystyle \textstyle {\frac {9}{10}}\log _{2}{10}\approx 2.99}$ ${\displaystyle 10^{k}}$ ${\displaystyle 10^{k+1}}$ ${\displaystyle k\in \mathbb {N} }$

• There are no single-digit Keith numbers. It is believed that there are others for which there are no -digit Keith numbers. ${\displaystyle 10}$ ${\displaystyle k}$ ${\displaystyle k}$
• Define a Keith cluster as a set of two or more Keith numbers with exactly the same number of places, where all Keith numbers are integral multiples of the first Keith number in this cluster. Only three such clusters are known:

${\displaystyle (14,28),(1104,2208)}$ and ${\displaystyle (31331,62662,93993)}$

Keith suspects that these three clusters are the only ones. But he admits that he has no idea how to prove that.

Keith Prime Numbers

A Keith number that is prime is called a Keith prime number .

Examples

• The smallest Keith prime numbers are the following:

19, 47, 61, 197, 1084051, 74596893730427,… (Follow A048970 in OEIS )

Generalizations

So far, only Keith numbers in the decimal system , i.e. the base, have been treated. For example, the Keith number would be the base number and with this base you would not have a Keith number (the corresponding sequence would be and you can see that there is no Keith number because it does not appear in the sequence). Therefore the respective base plays a big role in Keith numbers. ${\displaystyle b=10}$ ${\displaystyle n=47}$ ${\displaystyle b=8}$ ${\displaystyle n=47={\underline {5}}\cdot 8^{1}+{\underline {7}}\cdot 8^{0}=57_{8}}$ ${\displaystyle b=8}$ ${\displaystyle 5_{8},7_{8},14_{8},23_{8},37_{8},62_{8},\ldots }$ ${\displaystyle n=57_{8}}$

A Keith number for the base ${\displaystyle b}$ is a natural number that defines and is contained in a special mathematical sequence by means of its base digits . ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle b}$

Examples

• Let be a number in the duodecimal system , i.e. the base . Then you get the following sequence (for lack of further digits and ): ${\displaystyle n=49_{12}}$ ${\displaystyle b=12}$ ${\displaystyle A:=10}$ ${\displaystyle B:=11}$

${\displaystyle n=4_{12},9_{12},11_{12},1A_{12},2B_{12},49_{12},78_{12},\ldots }$

You can see that the number actually appears in the sequence. So a Keith number is the base . ${\displaystyle n=49_{12}}$ ${\displaystyle n=49_{12}}$ ${\displaystyle b=12}$

• The following numbers are the smallest Keith numbers for the base , i.e. in the duodecimal system: ${\displaystyle b=12}$

11, 15, 1B, 22, 2A, 31, 33, 44, 49, 55, 62, 66, 77, 88, 93, 99, AA, BB, 125, 215, 24A, 405, 42A, 654, 80A, 8A3, A59, 1022, 1662, 2044, 3066, 4088, 4A1A, 4AB1, 50AA, 8538, B18B, 17256, 18671, 24A78, 4718B, 517BA, 157617, 1A265A, 5A4074, 5AB140, 6B1449, 6B8515, ...

Inverted Keith numbers

Be a natural number with digits , so ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle d_{k-1},d_{k-2},\ldots ,d_{1},d_{0}}$

${\displaystyle n=\sum _{i=0}^{k-1}10^{i}{d_{i}}}$

Be a math sequence that starts with the values . Each further sequence element is the sum of the preceding sequence elements. If the number in this sequence is contained in the reverse order (i.e. with reversed digits), then a reverse Keith number (English reverse Keith number , but also revrepfigit number (short for rev erse rep licating Fi bonacci-like di git )) . Because single-digit numbers trivially satisfy this property, they are usually not accepted as reverse Keith numbers. So it has to be. It is not known whether there are infinitely many reversed Keith numbers. ${\displaystyle S_{n}}$ ${\displaystyle d_{k-1},d_{k-2},\ldots ,d_{1},d_{0}}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle S_{n}}$ ${\displaystyle n}$ ${\displaystyle n\geq 10}$

Examples

• Be the -digit number . Then the first members of the sequence are as follows: ${\displaystyle n}$ ${\displaystyle 3}$ ${\displaystyle n=341}$ ${\displaystyle s_{n}}$ ${\displaystyle S_{n}}$

3 , 4 , 1 , 8, 13, 22, 43, 78, 143 , 264, 485, 892, 1641, 3018, 5551, ...

The following term is the sum of the three preceding terms and . So it is . For example is . Because the -digit number is in this sequence and is exactly the reverse of the sequence of digits , is a reverse Keith number. ${\displaystyle s_{i}}$ ${\displaystyle s_{i-3},s_{i-2}}$ ${\displaystyle s_{i-1}}$ ${\displaystyle s_{i}=s_{i-3}+s_{i-2}+s_{i-1}}$ ${\displaystyle 485=78+143+264}$ ${\displaystyle 3}$ ${\displaystyle m=143}$ ${\displaystyle m=143}$ ${\displaystyle n=341}$ ${\displaystyle n=341}$

• The following numbers are the smallest inverted Keith numbers:

12, 24, 36, 48, 52, 71, 341, 682, 1285, 5532, 8166, 17593, 28421, 74733, 90711, 759664, 901921, 1593583, 4808691, 6615651, 6738984, 8366363, 8422611, 26435142, 54734431, 57133931, 79112422, 89681171, 351247542, 428899438, 489044741, 578989902, ... (sequence A097060 in OEIS )

Note that there are no reverse Keith numbers that end with a zero. These are not allowed, especially since these zeros, if you turn the digits of the number around, would be at the beginning and a zero is not allowed at the beginning.

• The following numbers are the smallest inverted Keith prime numbers :

71, 1593583, 54734431, ...

Individual evidence

1. ↑ Mike Keith: Repfigit Numbers . Ed .: J. Recr. Math. Band 19 , no. 2 , 1987, pp. 41-42 .
2. ↑ ^{a b c d} Mike Keith: Keith Numbers. Retrieved December 30, 2018 . 
3. ↑ ^{a b c d e f g} Eric W. Weisstein : Keith Number . In: MathWorld (English).
4. ↑ ^{a b} Jhon J. Bravo, Sergio Guzmán, Florian Luca: Repdigit Keith numbers. Lithuanian Mathematical Journal 53 (2), 2013, pp. 143–148 , accessed on December 30, 2018 (English). 

Web links

• Eric W. Weisstein : Keith Number . In: MathWorld (English).
• Eric W. Weisstein : Repfigit Number . In: MathWorld (English).
• Martin Klazar, Florian Luca: Counting Keith numbers. Journal of Integer Sequences 10 (2), 2007, pp. 1–10 , accessed on December 30, 2018 (English).