Editing Repdigit (section)

{{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>

However, 2323232323 is not a repdigit.

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>