Repunit

Repunit is a suitcase word made up of the English words repeated and unit and designates a number that only contains the number  1. A repunit is a special repdigit ("schnapps number"); the name Repunit was coined in 1966 by Albert H. Beiler . In German, the term Einserkolonne or Einserschlange is also used.

A prime repunit or repunit prime number is a repunit that is also a prime number .

definition

Mathematically, repunits (in the decimal system ) are defined as

${\displaystyle R_{n}={\frac {10^{n}-1}{9}}=\sum _{k=0}^{n-1}10^{k}=\overbrace {11\dotso 1} ^{n{\text{ Ziffern}}}}$, with . ${\displaystyle n\in \mathbb {N} ^{+}}$

A recursive definition can also be specified:

${\displaystyle R_{n}={\begin{cases}1,&{\text{wenn }}n=1{\text{,}}\\10^{n-1}+R_{n-1},&{\text{wenn }}n\geq 2{\text{.}}\end{cases}}}$

So the number consists of exactly ones ( ). The sequence of repunits begins as follows: 1, 11, 111, 1111, ... (sequence A002275 in OEIS ). ${\displaystyle R_{n}}$ ${\displaystyle n}$ ${\displaystyle n\geq 1}$

Repunit primes

The definition of repunits arose historically in the search for a decomposition of such numbers into their prime factors . The question of whether a repunit number is a prime number preoccupied mathematicians as early as the 19th century. For example, Carl Gustav Jacob Jacobi wrote a work entitled “ Investigating whether the number 11111111111 is a prime number or not. A curiosity caused by Dase . "

It is easy to show that is divisible by if is divisible by . For example, it is divisible by : 111111111 = 111 · 1001001. Therefore, it must be a prime number so that it can be a prime number. However, this condition is not sufficient, for example, no prime number is there . ${\displaystyle R_{n}}$ ${\displaystyle R_{a}}$ ${\displaystyle n}$ ${\displaystyle a}$ ${\displaystyle R_{9}}$ ${\displaystyle R_{3}}$ ${\displaystyle n}$ ${\displaystyle R_{n}}$ ${\displaystyle R_{3}}$ ${\displaystyle R_{3}=111=3\cdot 37}$

Except for this example of can only be a divisor of (for a prime number ) if for a specific one . ${\displaystyle R_{3}}$ ${\displaystyle p}$ ${\displaystyle R_{n}}$ ${\displaystyle n}$ ${\displaystyle p=2kn+1}$ ${\displaystyle k}$

Repunit primes are rare. is a prime number for (sequence A004023 in OEIS ). The Harvey Dubner and October 2000 by Lew Baxter found in September 1999 and are likely primes. At the end of March 2007, Paul Bourdelais and Harvey Dubner identified a suspect prime number, four months later Maksym Voznyy and Anton Budnyy found the currently (2013) largest known probable repunit prime number. It is believed that there are an infinite number of repunit primes. ${\displaystyle R_{n}}$ ${\displaystyle n=2,19,23,317,1031,\ldots }$ ${\displaystyle R_{49081}}$ ${\displaystyle R_{86453}}$ ${\displaystyle R_{109297}}$ ${\displaystyle R_{270343}}$

Generalized repunits

Since the above definition of repunits is based on the decimal system, this definition may seem arbitrary at first. However, one can generalize the underlying idea by defining repunits with respect to any base : ${\displaystyle b}$

${\displaystyle R_{n}^{(b)}={\frac {b^{n}-1}{b-1}}=\sum _{k=0}^{n-1}b^{k}=\underbrace {\overbrace {11\dotso 1} ^{n{\text{ Ziffern}}}{}_{b}} _{{\text{Zahl zur Basis }}b}}$With , , ${\displaystyle b,n\in \mathbb {N} }$ ${\displaystyle b\geq 2}$ ${\displaystyle n\geq 1}$

The generalized recursive definition is:

${\displaystyle R_{n}^{(b)}={\begin{cases}1,&{\text{wenn }}n=1{\text{,}}\\b^{n-1}+R_{n-1}^{(b)},&{\text{wenn }}n\geq 2{\text{.}}\end{cases}}}$

The number consists of exactly ones ( ) if it is noted as a number for the base ( regardless of the base it is always 1). ${\displaystyle R_{n}^{(b)}}$ ${\displaystyle n}$ ${\displaystyle n\geq 1}$ ${\displaystyle b}$ ${\displaystyle R_{1}^{(b)}}$

Table of values ​​for some repunits as an example:

Comparison of some repunit values for common number systems ${\displaystyle R_{n}^{(b)}}$

${\displaystyle R_{n}^{(b)}}$	Binary system ${\displaystyle b=2}$		Octal system ${\displaystyle b=8}$		Decimal system ${\displaystyle b=10}$	Hexadecimal system ${\displaystyle b=16}$
${\displaystyle n}$	binary	decimal	octal	decimal	decimal	hexadecimal	decimal
1	1 2	1 10	1 8	1 10	1 10	1 16	1 10
2	11 2	3 10	11 8	9 10	11 10	11 16	17 10
3	111 2	7 10	111 8	73 10	111 10	111 16	273 10
4th	1111 2	15 10	1111 8	585 10	1111 10	1111 16	4369 10
5	11111 2	31 10	11111 8	4681 10	11111 10	11111 16	69905 10
6th	111111 2	63 10	1111118	37449 10	111111 10	111111 16	1118481 10
7th	1111111 2	127 10	1111111 8	299593 10	1111111 10	1111111 16	17895697 10
8th	11111111 2	255 10	11111111 8	2396745 10	11111111 10	11111111 16	286331153 10
9	111111111 2	511 10	111111111 8	19173961 10	111111111 10	111111111 16	4581298449 10
10	1111111111 2	1023 10	1111111111 8	153391689 10	1111111111 10	1111111111 16	73300775185 10

It is easy to prove that for anything that is not divisible by 2 or without remainder , there is a repunit to the base that is a multiple of . ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle 2p}$ ${\displaystyle n}$

The base 2 repunits are known as the Mersenne numbers : ${\displaystyle M_{n}=2^{n}-1}$

The repunit prime numbers are a subset of the permutable prime numbers , i.e. the prime numbers that remain prime if you swap their digits at will.

Andy Steward calculated a particularly large generalized repunit prime number with 37,090 digits in 2006 . In 2010, Tom Wu found an even bigger one with 41,382 jobs. The currently (August 2020) largest known generalized repunit prime is with 95,202 digits. ${\displaystyle \textstyle {\frac {28839^{8317}-1}{28838}}}$ ${\displaystyle \textstyle {\frac {1549^{12973}-1}{1548}}}$ ${\displaystyle {\frac {7176^{24691}-1}{7175}}}$

Repunit prime numbers on different bases

Examples:

• The repunit is based on a prime number because it is a prime number. ${\displaystyle R_{7}^{(5)}=1111111_{5}}$ ${\displaystyle b=5}$ ${\displaystyle 1111111_{5}={\underline {1}}\cdot 5^{6}+{\underline {1}}\cdot 5^{5}+{\underline {1}}\cdot 5^{4}+{\underline {1}}\cdot 5^{3}+{\underline {1}}\cdot 5^{2}+{\underline {1}}\cdot 5^{1}+{\underline {1}}\cdot 5^{0}=15625+3125+625+125+25+5+1=19531\in \mathbb {P} }$
• The following is a table of the smallest repunit primes to bases , written in the decimal system ${\displaystyle R_{n}^{(b)}}$ ${\displaystyle b\leq 12}$

Base ${\displaystyle b}$	the smallest repunit primes to bases , written in the decimal system ${\displaystyle R_{n}^{(b)}={\overbrace {11\ldots 1} ^{n{\text{ Einser}}}}_{b}}$ ${\displaystyle b\leq 12}$	OEIS episode
	the associated , for which the above repunits are prime numbers ${\displaystyle n}$	OEIS sequence
2	3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, ... (all Mersenne prime numbers )	Follow A000668 in OEIS
	2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643609, ...	Follow A000043 in OEIS
3	13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013, ...	Follow A076481 in OEIS
	3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ...	Follow A028491 in OEIS
4th	5 (the only one because is and the number for odd is a factor of and for even is a factor of ) ${\displaystyle 4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)}$ ${\displaystyle 3}$ ${\displaystyle n}$ ${\displaystyle 2^{n}+1}$ ${\displaystyle n}$ ${\displaystyle 2^{n}-1}$
	2
5	31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, ...	Follow A086122 in OEIS
	3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ...	Follow A004061 in OEIS
6th	7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, ...	Follow A165210 in OEIS
	2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ...	Follow A004062 in OEIS
7th	2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457, 138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601, ...	Follow A102170 in OEIS
	5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...	Follow A004063 in OEIS
8th	73 (the only one because and the first factor is divisible by 7 if it is not divisible by 3 or the second factor is divisible by 7 if it is a multiple of 3) ${\displaystyle 8^{n}-1=\left(4^{n}+2^{n}+1\right)\left(2^{n}-1\right)}$ ${\displaystyle 4^{n}+2^{n}+1}$ ${\displaystyle n}$ ${\displaystyle 2^{n}-1}$ ${\displaystyle n}$
	3
9	there is not a single prime repunit with this base because and are both and even ${\displaystyle 9^{n}-1=\left(3^{n}+1\right)\left(3^{n}-1\right)}$ ${\displaystyle 3^{n}+1}$ ${\displaystyle 3^{n}-1}$
	-
10	11, 1111111111111111111, 11111111111111111111111, ...	Follow A004022 in OEIS
	2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...	Follow A004023 in OEIS
11	50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953, 56700023252179573962582828126754197 864 348 680538828324105 794 573 548 680 538 388 324 641 563 548 713 444 986 805 388 388 324 465 793 548 648 68053882174
	17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...	Follow A005808 in OEIS
12	13, 157, 22621, 29043636306420266077, 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801351464052482842970801320278910641601 4842970801351465052482842970801351465052482842970801351465052482842970801351467019641601 787185901320278910654
	2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...	Follow A004064 in OEIS

Web links

• Eric W. Weisstein : Repunit . In: MathWorld (English).
• Repunit prime numbers research project .

Individual evidence

1. ^ Albert H. Beiler: Recreations in the Theory of Numbers. The queen of mathematics entertains. 2nd Edition. Dover, New York 1966, chap. XI, p. 83 ff.
2. ^ Giovanni Di Maria: The Repunit Primes Project .
3. ^ Chris K. Caldwell: The Prime Glossery: Repunit .
4. ^ Andy Steward: Titanic Prime Generalized Repunits. ( Memento from October 19, 2013 in the Internet Archive )
5. ↑ Chris K. Caldwell: Generalized Repunit. Retrieved on August 21, 2020 (English).