In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary).
Repdigits are the representation in base of the number where is the repeated digit and is the number of repetitions. For example, the repdigit 77777 in base 10 is .
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 118, 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 11n − 1. 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, ... .
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 (or a "GET"), such as 22222222.
History
The concept of a repdigit has been studied under that name since at least 1974, and earlier called them "monodigit numbers". The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in Fortaleza at Brazil. The first problem in this competition, proposed by Mexico, was as follows: A number is called "Brazilian" if there exists an integer b such that 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.
Primes and repunits
For a repdigit to be 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. Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 1114 = 3 × 7 and 111 = 11110 = 3 × 37 are not prime. In any given base b, every repunit prime in that base with the exception of 11b (if it is prime) is a Brazilian prime. The smallest Brazilian primes are 7 = 1112, 13 = 1113, 31 = 111112 = 1115, 43 = 1116, 73 = 1118, 127 = 11111112, 157 = 11112, ...
While 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 . This convergence implies that the Brazilian primes form a vanishingly small fraction of all prime numbers. For instance, among the 3.7×1010 prime numbers below 1012, only 8.8×104 are Brazilian.
The decimal repunit primes have the form for the values of n listed in . It has been conjectured that there are infinitely many decimal repunit primes. The binary repunits are the Mersenne numbers and the binary repunit primes are the Mersenne primes.
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. 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, ...
If a Fermat number is prime, it is not Brazilian, but if it is composite, it is Brazilian. Contradicting a previous conjecture, Resta, Marcus, Grantham, and Graves found examples of Sophie Germain primes that are Brazilian, the first one is 28792661 = 1111173.