View source for Repdigit

{{Short description|Natural number with a decimal representation made of repeated instances of the same digit}}
In [[recreational mathematics]], a '''repdigit''' or sometimes '''monodigit'''<ref name=beiler>{{cite book|last1=Beiler|first1=Albert|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|url=https://archive.org/details/recreationsinthe0000beil|url-access=registration|date=1966|publisher=Dover Publications|location=New York|isbn=978-0-486-21096-4|page=[https://archive.org/details/recreationsinthe0000beil/page/83 83]|edition=2}}</ref> is a [[natural number]] composed of repeated instances of the same [[Numerical digit|digit]] in a [[positional number system]] (often implicitly [[decimal]]).  The word is a [[portmanteau]] of "repeated" and "digit".
Examples are [[11 (number)|11]], [[Number of the beast|666]], [[Four fours|4444]], and [[Six nines in pi|999999]]. All repdigits are [[palindromic number]]s and are multiples of [[repunit]]s. Other well-known repdigits include the [[repunit prime]]s and in particular the [[Mersenne prime]]s (which are repdigits when represented in binary).

Any such number can be represented as follows

 <math>
 \underbrace{n n \ldots nn}_{k } = \frac{(nn - n)^k - n^k}{(nn - 2\cdot n) \cdot n^{(k-2)}}
 </math>

Where nn is the concatenation of n with n. k the number of concatenated n. 

nn can be represented mathematically as

<math>
n\cdot\left(10^{\lfloor\log_{10}(n)\rfloor+1}+1\right)
</math>

for n = 23 and k = 5, the formula will look like this 

<math>
\frac{(2323 - 23)^5 - 23^5}{(2323 - 2\cdot 23) \cdot 23^{(5-2)}} = \frac{64363429993563657}{27704259} = \underbrace{2323232323}_{5}
</math>

Also, any number can be decomposed into the sum and difference of the repdigit numbers.

For example 3453455634 = 3333333333 + (111111111 + (9999999 - (999999 - (11111 + (77 + (2))))))

Repdigits are the representation in [[radix|base]] <math>B</math> of the number <math>x\frac{B^y -1}{B-1}</math> where <math>0<x<B</math> is the repeated digit and <math>1<y</math> is the number of repetitions. For example, the repdigit 77777 in base 10 is <math>7\times\frac{10^5-1}{10-1}</math>.

A variation of repdigits called '''Brazilian numbers''' are numbers that can be written as a repdigit in some base, not allowing the repdigit 11, and not allowing the single-digit numbers (or all numbers will be Brazilian). For example, 27 is a Brazilian number because 27 is the repdigit 33 in base 8, while 9 is not a Brazilian number because its only repdigit representation is 11<sub>8</sub>, not allowed in the definition of Brazilian numbers. The representations of the form 11 are considered trivial and are disallowed in the definition of Brazilian numbers, because all natural numbers ''n'' greater than two have the representation 11<sub>''n'' &minus; 1</sub>.<ref name=schott>{{cite journal|last=Schott|first=Bernard|date=March 2010|doi=10.1051/quadrature/2010005|issue=76|journal=Quadrature|pages=30–38|title=Les nombres brésiliens|url=https://oeis.org/A125134/a125134.pdf|language=fr}}</ref> The first twenty Brazilian numbers are
: 7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, ... {{OEIS|id=A125134}}.

On some websites (including [[imageboards]] like [[4chan]]), it is considered an auspicious event when the sequentially-assigned ID number of a post is a repdigit, such as 22,222,222, which is one type of "GET"{{Clarify|reason=what is a "GET"? link or clarify, not mentioned in the body of the article either|date=December 2024}} (others including round numbers like 34,000,000, or sequential digits like 12,345,678).<ref name="faqget">{{cite web |url=https://www.4chan.org/faq.php#get |title=FAQ on GETs |website=4chan |access-date=March 14, 2007}}</ref><ref name="convo">{{Cite web|url=http://theconversation.com/how-an-ancient-egyptian-god-spurred-the-rise-of-trump-72598|title=How an ancient Egyptian god spurred the rise of Trump|first1=Adrià Salvador|last1=Palau|first2=Jon|last2=Roozenbeek|date=March 7, 2017|website=The Conversation}}</ref>

==History==
The concept of a repdigit has been studied under that name since at least 1974,<ref>{{cite journal|last=Trigg|first=Charles W.|author-link=Charles W. Trigg|journal=The Fibonacci Quarterly|mr=354535|pages=209–212|title=Infinite sequences of palindromic triangular numbers|url=https://www.mathstat.dal.ca/FQ/Scanned/12-2/trigg.pdf|volume=12|year=1974|issue=2 |doi=10.1080/00150517.1974.12430760 }}</ref> and earlier {{harvtxt|Beiler|1966}} called them "monodigit numbers".<ref name=beiler/> The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in [[Fortaleza]], Brazil. The first problem in this competition, proposed by Mexico, was as follows:<ref>{{cite book|title=Hypermath|author=Pierre Bornsztein| location=Paris|publisher=Vuibert| page=7, exercice a35<!--endif p.totales-->|year=2001}}</ref> 
<blockquote>
A number {{nowrap|''n'' > 0}} is called "Brazilian" if there exists an integer ''b'' such that {{nowrap|1 <  ''b'' < ''n'' – 1}} for which the representation of ''n'' in base ''b'' is written with all equal digits. Prove that 1994 is Brazilian and that 1993 is not Brazilian.
</blockquote>

==Primes and repunits==
{{main|Repunit prime}}
For a repdigit to be [[prime number|prime]], it must be a [[repunit]] (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7. In particular, as Brazilian repunits do not allow the number of digits to be exactly two, Brazilian primes must have an odd prime number of digits.{{sfnp|Schott|2010|loc=Theorem 2}} Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 111<sub>4</sub> = 3 × 7 and 111 = 111<sub>10</sub> = 3 × 37 are not prime. In any given base ''b'', every repunit prime in that base with the exception of 11<sub>''b''</sub> (if it is prime) is a Brazilian prime. The smallest Brazilian primes are
:7 = 111<sub>2</sub>, 13 = 111<sub>3</sub>, 31 = 11111<sub>2</sub> = 111<sub>5</sub>, 43 = 111<sub>6</sub>, 73 = 111<sub>8</sub>, 127 = 1111111<sub>2</sub>, 157 = 111<sub>12</sub>, ... {{OEIS|id=A085104}}

While [[Divergence of the sum of the reciprocals of the primes|the sum of the reciprocals of the prime numbers]] is a divergent series, the sum of the reciprocals of the Brazilian prime numbers is a convergent series whose value, called the "Brazilian primes constant", is slightly larger than 0.33 {{OEIS|id=A306759}}.{{sfnp|Schott|2010|loc=Theorem 4}} This convergence implies that the Brazilian primes form a vanishingly small fraction of all prime numbers. For instance, among the 3.7&times;10<sup>10</sup> prime numbers smaller than 10<sup>12</sup>, only 8.8&times;10<sup>4</sup> are Brazilian.

The [[decimal]] repunit primes have the form <math>R_n=\tfrac{10^n-1}9\ \mbox{with } n\ge3</math> for the values of ''n'' listed in {{OEIS2C|id=A004023}}. It has been conjectured that there are infinitely many decimal repunit primes.<ref>Chris Caldwell, "[http://primes.utm.edu/glossary/page.php?sort=Repunit The Prime Glossary: repunit]" at The [[Prime Pages]]</ref> The [[binary number|binary]] repunits are the [[Mersenne number]]s and the binary repunit primes are the [[Mersenne prime]]s.

It is unknown whether there are infinitely many Brazilian primes. If the [[Bateman–Horn conjecture]] is true, then for every prime number of digits there would exist infinitely many repunit primes with that number of digits (and consequentially infinitely many Brazilian primes). Alternatively, if there are infinitely many decimal repunit primes, or infinitely many Mersenne primes, then there are infinitely many Brazilian primes.{{sfnp|Schott|2010|loc=Sections V.1 and V.2}} Because a vanishingly small fraction of primes are Brazilian, there are infinitely many non-Brazilian primes, forming the sequence
:2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, ... {{OEIS|id=A220627}}

If a [[Fermat number]] <math>F_n = 2^{2^n} + 1</math> is prime, it is not Brazilian, but if it is composite, it is Brazilian.{{sfnp|Schott|2010|loc=Proposition 3}}
Contradicting a previous conjecture,{{sfnp|Schott|2010|loc=Conjecture 1}} Resta, Marcus, Grantham, and Graves found examples of [[Sophie Germain prime]]s that are Brazilian, the first one is 28792661 = 11111<sub>73</sub>.<ref>{{cite arXiv|title=Brazilian primes which are also Sophie Germain primes|first1=Jon|last1=Grantham|first2=Hester|last2=Graves|year=2019|class=math.NT|eprint=1903.04577}}</ref>

==Non-Brazilian composites and repunit powers==
The only positive integers that can be non-Brazilian are 1, 6, the [[prime number|prime]]s, and the [[square number|square]]s of the primes, for every other number is the product of two factors ''x'' and ''y'' with 1 < ''x'' < ''y'' &minus; 1, and can be written as ''xx'' in base ''y'' &minus; 1.{{sfnp|Schott|2010|loc=Theorem 1}} If a square of a prime ''p''<sup>2</sup> is Brazilian, then prime ''p'' must satisfy the [[Diophantine equation]] <div style="text-align: center;">''p''<sup>2</sup> = 1 + ''b'' + ''b''<sup>2</sup> + ... + ''b''<sup>''q''-1</sup> with ''p'', ''q'' ≥ 3 primes and ''b'' >= 2.</div>
Norwegian mathematician [[Trygve Nagell]] has proved<ref>{{cite journal|last=Nagell|first=Trygve|title=Sur l'équation indéterminée (x<sup>n</sup>-1)/(x-1) = y |journal=Norsk Matematisk Forenings Skrifter|year=1921|volume=3|issue=1|pages=17–18}}.</ref> that this equation has only one solution when ''p'' is prime corresponding to {{nowrap|1=(''p'', ''b'', ''q'') = (11, 3, 5)}}. Therefore, the only squared prime that is Brazilian is 11<sup>2</sup> = 121 = 11111<sub>3</sub>.
There is also one more nontrivial repunit square, the solution (''p'', ''b'', ''q'') = (20, 7, 4) corresponding to 20<sup>2</sup> = 400 = 1111<sub>7</sub>, but it is not exceptional with respect to the classification of Brazilian numbers because 20 is not prime.

[[Perfect power]]s that are repunits with three digits or more in some base ''b'' are described by the [[Diophantine equation]] of Nagell and [[Wilhelm Ljunggren|Ljunggren]]<ref>{{cite journal|language=no|last=Ljunggren|first=Wilhelm|title=Noen setninger om ubestemte likninger av formen (x<sup>n</sup>-1)/(x-1) = y<sup>q</sup>|journal=Norsk Matematisk Tidsskrift|year=1943|volume=25|pages=17–20}}.</ref><div style="text-align: center;">''n''<sup>''t''</sup> = 1 + ''b'' + ''b''<sup>2</sup> +...+ ''b''<sup>''q''-1</sup> with ''b, n, t'' > 1 and ''q'' > 2.</div>
Yann Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits. They are 121, 343, and 400 {{OEIS|A208242}}, the two squares listed above and the cube 343 = 7<sup>3</sup> = 111<sub>18</sub>.<ref>{{cite journal|last1=Bugeaud|first1=Yann |last2=Mignotte|first2=Maurice|title=L'équation de Nagell-Ljunggren (x<sup>n</sup>-1)/(x-1) = y<sup>q</sup>|journal=L'Enseignement Mathématique|year=2002|volume=48|pages=147–168|url=https://www.e-periodica.ch/digbib/view?pid=ens-001:2002:48#261}}.</ref>

==''k''-Brazilian numbers==
* The number of ways such that a number ''n'' is Brazilian is in {{OEIS2C|id=A220136}}. Hence, there exist numbers that are non-Brazilian and others that are Brazilian; among these last integers, some are once Brazilian, others are twice Brazilian, or three times, or more. A number that is ''k'' times Brazilian is called ''k-Brazilian number''.
*Non-Brazilian numbers or 0''-Brazilian numbers'' are constituted with 1 and 6, together with some primes and some squares of primes. The sequence of the non-Brazilian numbers begins with 1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, ... {{OEIS|id=A220570}}.
* The sequence of 1''-Brazilian numbers'' is composed of other primes, the only square of prime that is Brazilian, 121, and composite numbers {{nowrap|≥ 8}} that are the product of only two distinct factors such that {{nowrap|1=''n'' = ''a'' × ''b'' = ''aa''<sub>''b''–1</sub>}} with {{nowrap|1 < ''a'' < ''b'' – 1}}.  {{OEIS|id=A288783}}.
* The 2''-Brazilian numbers'' {{OEIS|id=A290015}} consists of composites and only two primes: 31 and 8191. Indeed, according to [[Goormaghtigh conjecture]], these two primes are the only known solutions of the [[Diophantine equation]]: <div style="text-align: center;"><math>p=\frac{x^m - 1}{x-1}=\frac{y^n - 1}{y - 1}</math> with ''x'', ''y''&nbsp;>&nbsp;1 and ''n'', ''m''&nbsp;>&nbsp;2 :</div>  
**(''p'',&nbsp;''x'',&nbsp;''y'',&nbsp;''m'',&nbsp;''n'')&nbsp;=&nbsp;(31,&nbsp;5,&nbsp;2,&nbsp;3,&nbsp;5) corresponding to 31 = 11111<sub>2</sub> = 111<sub>5</sub>, and,
**(''p'',&nbsp;''x'',&nbsp;''y'',&nbsp;''m'',&nbsp;''n'')&nbsp;=&nbsp;(8191,&nbsp;90,&nbsp;2,&nbsp;3,&nbsp;13) corresponding to 8191 = 1111111111111<sub>2</sub> = 111<sub>90</sub>, with 11111111111 is the [[repunit]] with thirteen digits 1.
* For each sequence of ''k-Brazilian numbers'', there exists a smallest term. The sequence with these smallest ''k''-Brazilian numbers begins with 1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, ... and are in {{OEIS2C|id=A284758}}. For instance, 40 is the smallest ''4-Brazilian number'' with  40 = 1111<sub>3</sub> = 55<sub>7</sub> = 44<sub>9</sub> = 22<sub>19</sub>.
* In the ''Dictionnaire de (presque) tous les nombres entiers'',<ref>{{cite book|title=Dictionnaire de (presque) tous les nombres entiers  |author=Daniel Lignon|location=Paris|publisher=Ellipses|pages=420<!--endif p.totales-->|year=2012}}</ref> Daniel Lignon proposes that an integer is ''highly Brazilian'' if it is a positive integer with more Brazilian representations  than any smaller positive integer has. This definition comes from the definition of [[highly composite number]]s created by [[Srinivasa Ramanujan]] in 1915. The first numbers ''highly Brazilian'' are  1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, ... and are exactly in {{OEIS2C|id=A329383}}. From 360 to 321253732800 (maybe more), there are 80 successive [[highly composite number]]s that are also highly Brazilian numbers, see  {{OEIS2C|id=A279930}}.

==Numerology==
Some popular media publications have published articles suggesting that repunit numbers have [[numerological]] significance, describing them as "[[angel numbers]]".<ref>{{Cite magazine |date=2023-06-29 |title=The 333 angel number is very powerful in numerology – here's what it means |url=https://www.glamourmagazine.co.uk/article/333-angel-number-meaning |access-date=2023-08-28 |magazine=Glamour UK |language=en-GB}}</ref><ref>{{Cite magazine |date=24 December 2021 |title=Everything You Need to Know About Angel Numbers |url=https://www.allure.com/story/what-are-angel-numbers |access-date=28 August 2023 |magazine=Allure |language=en-US}}</ref><ref>{{Cite magazine |date=21 July 2021 |title=Everything You Need to Know About Angel Numbers |url=https://www.cosmopolitan.com/lifestyle/a37079416/angel-numbers-numerology/ |access-date=2023-08-28 |magazine=Cosmopolitan |language=en-US}}</ref>

==See also==
* [[Six nines in pi]]

==References==
{{Reflist}}

==External links==
* {{MathWorld|urlname=Repdigit|title=Repdigit}}
* [https://www.oei.es/historico/oim/ixoim.htm Problemas IX Olimpíada Iberoamericana de Matemática]

{{Wiktionary}}
{{Classes of natural numbers}}

[[Category:Base-dependent integer sequences]]