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In
recreational mathematics Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, a repdigit or sometimes monodigit is a
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
composed of repeated instances of the same digit in a positional number system (often implicitly
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
). The word is a
portmanteau In linguistics, a blend—also known as a blend word, lexical blend, or portmanteau—is a word formed by combining the meanings, and parts of the sounds, of two or more words together.
of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are
palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...
s and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s (which are repdigits when represented in binary). Any such number can be represented as follows \underbrace_ = \frac Where nn is the concatenation of n with n. k the number of concatenated n. nn can be represented mathematically as n\cdot\left(10^+1\right) for n = 23 and k = 5, the formula will look like this \frac = \frac = \underbrace_ 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 base B of the number x\frac where 0 is the repeated digit and 1 is the number of repetitions. For example, the repdigit 77777 in base 10 is 7\times\frac. 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 11''n'' − 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 An imageboard is a type of Internet forum that focuses on the posting of images, often alongside text and discussion. The first imageboards were created in Japan as an extension of the textboard concept. These sites later inspired the creation of ...
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" (others including round numbers like 34,000,000, or sequential digits like 12,345,678).


History

The concept of a repdigit has been studied under that name since at least 1974; earlier called them "monodigit numbers". The Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in
Fortaleza Fortaleza ( ; ; ) is the state capital of Ceará, located in Northeast Region, Brazil, Northeastern Brazil. It is Brazil's 4th largest city—Fortaleza surpassed Salvador, Bahia, Salvador in 2022 census with a population of slightly over 2.4 mi ...
, 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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, 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 11''b'' (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 smaller than 1012, only 8.8×104 are Brazilian. The
decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
repunit primes have the form R_n=\tfrac9\ \mbox n\ge3 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 number In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...
s and the binary repunit primes are the
Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s. 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 In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have studied them, is a natural number, positive integer of the form:F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers ...
F_n = 2^ + 1 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 prime In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 +&nbs ...
s that are Brazilian, the first one is 28792661 = 1111173.


Non-Brazilian composites and repunit powers

The only positive integers that can be non-Brazilian are 1, 6, the
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s, and the
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s of the primes, for every other number is the product of two factors ''x'' and ''y'' with 1 < ''x'' < ''y'' − 1, and can be written as ''xx'' in base ''y'' − 1. If a square of a prime ''p''2 is Brazilian, then prime ''p'' must satisfy the
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
''p''2 = 1 + ''b'' + ''b''2 + ... + ''b''''q''-1 with ''p'', ''q'' ≥ 3 primes and ''b'' >= 2.
Norwegian mathematician
Trygve Nagell Trygve Nagell or Trygve Nagel (13 July 1895 – 24 January 1988) was a Norwegian mathematician, known for his works on Diophantine equations in number theory. He was born in Oslo and died in Uppsala. Education and career He was born Nag ...
has proved that this equation has only one solution when ''p'' is prime corresponding to . Therefore, the only squared prime that is Brazilian is 112 = 121 = 111113. 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 ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
of Nagell and Ljunggren
''n''''t'' = 1 + ''b'' + ''b''2 +...+ ''b''''q''-1 with ''b, n, t'' > 1 and ''q'' > 2.
Yann Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits. They are 121, 343, and 400 , the two squares listed above and the cube 343 = 73 = 11118.


''k''-Brazilian numbers

* The number of ways such that a number ''n'' is Brazilian is in . Hence, there exist numbers that are non-Brazilian and others that are Brazilian; among these last integers, some are once Brazilian, others are twice Brazilian, or three times, or more. A number that is ''k'' times Brazilian is called ''k-Brazilian number''. *Non-Brazilian numbers or 0''-Brazilian numbers'' are constituted with 1 and 6, together with some primes and some squares of primes. The sequence of the non-Brazilian numbers begins with 1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, ... . * The sequence of 1''-Brazilian numbers'' is composed of other primes, the only square of prime that is Brazilian, 121, and composite numbers that are the product of only two distinct factors such that with . . * The 2''-Brazilian numbers'' consists of composites and only two primes: 31 and 8191. Indeed, according to Goormaghtigh conjecture, these two primes are the only known solutions of the
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
:
p=\frac=\frac with ''x'', ''y'' > 1 and ''n'', ''m'' > 2 :
**(''p'', ''x'', ''y'', ''m'', ''n'') = (31, 5, 2, 3, 5) corresponding to 31 = 111112 = 1115, and, **(''p'', ''x'', ''y'', ''m'', ''n'') = (8191, 90, 2, 3, 13) corresponding to 8191 = 11111111111112 = 11190, with 11111111111 is the repunit with thirteen digits 1. * For each sequence of ''k-Brazilian numbers'', there exists a smallest term. The sequence with these smallest ''k''-Brazilian numbers begins with 1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, ... and are in . For instance, 40 is the smallest ''4-Brazilian number'' with 40 = 11113 = 557 = 449 = 2219. * In the ''Dictionnaire de (presque) tous les nombres entiers'', Daniel Lignon proposes that an integer is ''highly Brazilian'' if it is a positive integer with more Brazilian representations than any smaller positive integer has. This definition comes from the definition of
highly composite number A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(' ...
s created by
Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...
in 1915. The first numbers ''highly Brazilian'' are 1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, ... and are exactly in . From 360 to 321253732800 (maybe more), there are 80 successive
highly composite number A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(' ...
s that are also highly Brazilian numbers, see .


Numerology

Some popular media publications have published articles suggesting that repunit numbers have numerological significance, describing them as " angel numbers".


See also

* Six nines in pi


References


External links

*
Problemas IX Olimpíada Iberoamericana de Matemática
{{Classes of natural numbers Base-dependent integer sequences