Brun's theorem
Encyclopedia
In number theory
, Brun's theorem was proved by Viggo Brun
in 1919. It states that the sum of the reciprocals of the twin prime
s (pairs of prime number
s which differ by 2) is convergent with a finite value known as Brun's constant, usually denoted by B2 . It has historical importance in the introduction of sieve methods.
Let
denote the number of primes
p ≤ x for which p + 2 is also prime (i.e.
is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we have
for some positive constant c.
This result shows that the sum of the reciprocals of the twin primes converges; in other words the p involved are a small set. In explicit terms the sum
either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant. Unlike the case for all prime numbers, we cannot conclude from this result that there is an infinite number of twin primes.
along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6 as of 18 January 2010 but this is not the largest computation of its type.
In 2002 Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate:
It is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Richard Crandall
and Carl Pomerance
reported that it is known rigorously that 1.83 < B2 < 2.347. Dominic Klyve showed conditionally that B2 < 2.1754 (assuming the extended Riemann hypothesis).
The digits of Brun's constant was used in a bid of $1,902,160,540 in the Nortel
patent auction. The bid was posted by Google
and was one of three Google bids based on mathematical constants.
There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:
with value:
This constant should not be confused with the Brun's constant for cousin prime
s, prime pairs of the form (p, p + 4), which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n.
. Then it is conjectured that
In particular,
for every
and all sufficiently large x.
Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large x,
where 4.5 corresponds to
in the above.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, Brun's theorem was proved by Viggo Brun
Viggo Brun
Viggo Brun was a Norwegian mathematician.He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the Technical University in Trondheim and in 1946 a professor at the University of Oslo...
in 1919. It states that the sum of the reciprocals of the twin prime
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...
s (pairs of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s which differ by 2) is convergent with a finite value known as Brun's constant, usually denoted by B2 . It has historical importance in the introduction of sieve methods.
Let
denote the number of primesPrime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
p ≤ x for which p + 2 is also prime (i.e.
is the number of twin primes with the smaller at most x). Then, for x ≥ 3, we havefor some positive constant c.
This result shows that the sum of the reciprocals of the twin primes converges; in other words the p involved are a small set. In explicit terms the sum
either has finitely many terms or has infinitely many terms but is convergent: its value is known as Brun's constant. Unlike the case for all prime numbers, we cannot conclude from this result that there is an infinite number of twin primes.
Brun's constant
By calculating the twin primes up to 1014 (and discovering the Pentium FDIV bugPentium FDIV bug
The Pentium FDIV bug was a bug in the Intel P5 Pentium floating point unit . Certain floating point division operations performed with these processors would produce incorrect results...
along the way), Thomas R. Nicely heuristically estimated Brun's constant to be 1.902160578. Nicely has extended his computation to 1.6 as of 18 January 2010 but this is not the largest computation of its type.
In 2002 Pascal Sebah and Patrick Demichel used all twin primes up to 1016 to give the estimate:
- B2 ≈ 1.902160583104.
It is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Richard Crandall
Richard Crandall
Richard E. Crandall is an American Physicist and computer scientist who has made contributions to computational number theory.He is most notable for the development of the irrational base discrete weighted transform, an important method of finding very large primes. He has, at various times, been...
and Carl Pomerance
Carl Pomerance
Carl Bernard Pomerance is a well-known number theorist. He attended college at Brown University and later received his Ph.D. from Harvard University in 1972 with a dissertation proving that any odd perfect number has at least 7 distinct prime factors. He immediately joined the faculty at the...
reported that it is known rigorously that 1.83 < B2 < 2.347. Dominic Klyve showed conditionally that B2 < 2.1754 (assuming the extended Riemann hypothesis).
The digits of Brun's constant was used in a bid of $1,902,160,540 in the Nortel
Nortel
Nortel Networks Corporation, formerly known as Northern Telecom Limited and sometimes known simply as Nortel, was a multinational telecommunications equipment manufacturer headquartered in Mississauga, Ontario, Canada...
patent auction. The bid was posted by Google
Google
Google Inc. is an American multinational public corporation invested in Internet search, cloud computing, and advertising technologies. Google hosts and develops a number of Internet-based services and products, and generates profit primarily from advertising through its AdWords program...
and was one of three Google bids based on mathematical constants.
There is also a Brun's constant for prime quadruplets. A prime quadruplet is a pair of two twin prime pairs, separated by a distance of 4 (the smallest possible distance). The first prime quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), (101, 103, 107, 109). Brun's constant for prime quadruplets, denoted by B4, is the sum of the reciprocals of all prime quadruplets:
with value:
- B4 = 0.87058 83800 ± 0.00000 00005, the error range having a 99% confidence level according to Nicely.
This constant should not be confused with the Brun's constant for cousin prime
Cousin prime
In mathematics, cousin primes are prime numbers that differ by four; compare this with twin primes, pairs of prime numbers that differ by two, and sexy primes, pairs of prime numbers that differ by six....
s, prime pairs of the form (p, p + 4), which is also written as B4. Wolf derived an estimate for the Brun-type sums Bn of 4/n.
Further results
Let . Then it is conjectured thatIn particular,
for every
and all sufficiently large x.Many special cases of the above have been proved. Most recently, Jie Wu proved that for sufficiently large x,
where 4.5 corresponds to
in the above.See also
- Proof that the sum of the reciprocals of the primes diverges
- Meissel–Mertens constant
External links
- Sebah, Pascal and Xavier Gourdon, Introduction to twin primes and Brun's constant computation, 2002. A modern detailed examination.
- Wolf's article on Brun-type sums