In recreational mathematics, a repdigit or sometimes monodigit[1] is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal).
All repdigits are palindromic numbers and are multiples of repunits.
For example 3453455634 = 3333333333 + (111111111 + (9999999 - (999999 - (11111 + (77 + (2)))))) Repdigits are the representation in [[radix|base]] of the number
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.
[2] The first twenty Brazilian numbers are 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"[clarification needed] (others including round numbers like 34,000,000, or sequential digits like 12,345,678).
[3][4] The concept of a repdigit has been studied under that name since at least 1974,[5] and earlier Beiler (1966) called them "monodigit numbers".
[1] 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:[6] 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.
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.
The smallest Brazilian primes are 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 (sequence A306759 in the OEIS).
For instance, among the 3.7×1010 prime numbers smaller than 1012, only 8.8×104 are Brazilian.
It has been conjectured that there are infinitely many decimal repunit primes.
[11] Contradicting a previous conjecture,[12] Resta, Marcus, Grantham, and Graves found examples of Sophie Germain primes that are Brazilian, the first one is 28792661 = 1111173.
There is also one more nontrivial repunit square, the solution (p, b, q) = (20, 7, 4) corresponding to 202 = 400 = 11117, but it is not exceptional with respect to the classification of Brazilian numbers because 20 is not prime.
Perfect powers that are repunits with three digits or more in some base b are described by the Diophantine equation of Nagell and Ljunggren[16]Yann Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits.
They are 121, 343, and 400 (sequence A208242 in the OEIS), the two squares listed above and the cube 343 = 73 = 11118.
[17] Some popular media publications have published articles suggesting that repunit numbers have numerological significance, describing them as "angel numbers".