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 those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ... composed of repeated instances of the same
digit in a
positional number system
Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ... (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 of the Hindu–Arabic numeral ... ). The word is a
portmanteau
A portmanteau word, or portmanteau (, ) is a blend of words[11, ](_blank)666
666 may refer to:
* 666 (number)
* 666 BC, a year
* AD 666, a year
* The number of the beast, a reference in the Book of Revelation in the New Testament
Places
* 666 Desdemona, a minor planet in the asteroid belt
* U.S. Route 666, an America ... 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 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ... repunit s. Other well-known repdigits include the repunit prime
In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book ''Recre ... 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 17 ... 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 . 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
Fortaleza (, locally , Portuguese for ''Fortress'') is the state capital of Ceará, located in Northeastern Brazil. It belongs to the Metropolitan mesoregion of Fortaleza and microregion of Fortaleza. It is Brazil's 5th largest city and the t ...
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 ... 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 below 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 of the Hindu–Arabic numeral ... 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 17th ... 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 17 ... Bateman–Horn conjecture In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A. Horn who proposed it in 1962. It provi ... Fermat number 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 + ... 73 .
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 ... square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ... 2 is Brazilian, then prime ''p'' must satisfy the Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ... ''p''2 = 1 + ''b'' + ''b''2 + ... + ''b''''q''-1 with ''p'', ''q'' ≥ 3 primes and ''b'' >= 2.
Norwegian mathematician Trygve Nagell 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 power s that are repunits with three digits or more in some base ''b'' are described by the Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ... 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
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ... 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'',highly composite number s created by Srinivasa Ramanujan
Srinivasa Ramanujan (; born Srinivasa Ramanujan Aiyangar, ; 22 December 188726 April 1920) was an Indian mathematician. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, ... highly composite number s that are also highly Brazilian numbers, see .
References
External links
*Problemas IX Olimpíada Iberoamericana de Matemática Base-dependent integer sequences
Recreational mathematics
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