# Repdigit

(Redirected from Brazilian number)

In recreational math, a repdigit or a monodigit[1] is a type of natural number. It is only made of the same repeated digit. The word is a portmanteau of repeated and digit. Examples can be numbers like 11, 666, 4444, and 999999. Repdigits are palindromic numbers (read the same forwards and backwards) and are multiples of repunits (a number that only has 1 in it).

Repdigits are the written in base ${\displaystyle B}$ of the number ${\displaystyle x{\frac {B^{y}-1}{B-1}}}$ where x is the digit that is repeated (${\displaystyle 0), and ${\displaystyle 1 is how many times that number repeats. For example, the repdigit 77777 in base 10 is ${\displaystyle 7\times {\frac {10^{5}-1}{10-1}}}$.

Brazilian numbers are another way of making repdigits. These are numbers that can be written as a repdigit in any base. A Brazilian Number can't be the repdigit 11 and it can't be a number with only one digit. 27 would be a Brazilian number because 27 is the repdigit 33 in base 8. 9 is not a Brazilian number because its only repdigit is 118.[2] 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, ... (sequence A125134 in the OEIS).

## History

Repdigits were studied since 1974.[3] In Beiler (1966), it was originally called "monodigit numbers".[1] 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:[4]

"A number n > 0 is called "Brazilian" if there exists an integer b such that 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."

## References

1. Beiler, Albert (1966). Recreations in the Theory of Numbers: The Queen of Mathematics Entertains (2 ed.). New York: Dover Publications. p. 83. ISBN 978-0-486-21096-4.
2. Schott, Bernard (March 2010). "Les nombres brésiliens" (PDF). Quadrature (in French) (76): 30–38. doi:10.1051/quadrature/2010005.
3. Trigg, Charles W. (1974). "Infinite sequences of palindromic triangular numbers" (PDF). The Fibonacci Quarterly. 12: 209–212. MR 0354535.
4. Pierre Bornsztein (2001). Hypermath. Paris. Vuibert. p. 7, exercice a35.