View source for Repdigit

In [[recreational math]], a '''repdigit''' or a '''monodigit'''<ref name="beiler">{{cite book|url=https://archive.org/details/recreationsinthe0000beil|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|last1=Beiler|first1=Albert|date=1966|publisher=Dover Publications|isbn=978-0-486-21096-4|edition=2|location=New York|page=[https://archive.org/details/recreationsinthe0000beil/page/83 83]|url-access=registration}}</ref> is a type of [[natural number]]. It is only made of the same repeated [[Numerical digit|digit]]. The word is a [[portmanteau]] of '''rep'''eated and '''digit'''. Examples can be numbers like [[11 (number)|11]], [[666 (number)|666]], 4444, and 999999. Repdigits are palindromic numbers (read the same forwards and backwards) and are [[Multiple|multiples]] of repunits (a number that only has 1 in it).

Repdigits are the written in [[Radix|base]] <math>B</math> of the number <math>x\frac{B^y -1}{B-1}</math> where ''x'' is the [[digit]] that is repeated (<math>0<x<B</math>), and <math>1<y</math> is how many times that number repeats. For example, the repdigit 77777 in base 10 is <math>7\times\frac{10^5-1}{10-1}</math>.

'''Brazilian numbers''' are another way of making repdigits. These are numbers that can be written as a repdigit in any [[Base (mathematics)|base]]. A Brazilian Number can't be the repdigit 11 and it can't be a number with only one [[digit]]. [[27 (number)|27]] would be a Brazilian number because 27 is the repdigit 33 in [[base 8]]. [[9 (number)|9]] is not a Brazilian number because its only repdigit is 11<sub>8</sub>.<ref name="schott">{{cite journal|last=Schott|first=Bernard|date=March 2010|title=Les nombres brésiliens|url=https://oeis.org/A125134/a125134.pdf|journal=Quadrature|language=fr|issue=76|pages=30–38|doi=10.1051/quadrature/2010005}}</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}}.

== History ==
Repdigits were studied since 1974.<ref>{{cite journal|last=Trigg|first=Charles W.|year=1974|title=Infinite sequences of palindromic triangular numbers|url=https://www.mathstat.dal.ca/FQ/Scanned/12-2/trigg.pdf|journal=The Fibonacci Quarterly|volume=12|pages=209–212|mr=354535}}</ref> In {{harvtxt|Beiler|1966}}, it was originally called "monodigit numbers".<ref name="beiler" /> Brazilian numbers were created later in 1994 in the 9th Iberoamerican Mathematical Olympiad. It took place in [[Fortaleza]], Brazil. The first problem in this [[competition]], proposed by Mexico, was this:<ref>{{cite book|title=Hypermath|author=Pierre Bornsztein|year=2001|location=Paris|page=7, exercice a35<!--endif p.totales-->|agency=Vuibert}}</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>

== Related pages ==
*[[Repunit]]

== References ==
{{Reflist}}

== Other websites ==

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

{{Classes of natural numbers}}

[[Category:Integer sequences]]