In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
, Brun's theorem states that the sum of the
reciprocals of the
twin prime
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pr ...
s (pairs of
prime number
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 which differ by 2)
converges to a finite value known as Brun's constant, usually denoted by ''B''
2 . Brun's theorem was proved by
Viggo Brun in 1919, and it has historical importance in the introduction of
sieve methods.
Asymptotic bounds on twin primes
The convergence of the sum of reciprocals of twin primes follows from bounds on the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
of the sequence of twin primes.
Let
denote the number of
primes
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 ...
''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
:
That is, twin primes are less frequent than prime numbers by nearly a logarithmic factor.
It follows from this bound that the sum of the reciprocals of the twin primes converges, or stated in other words, the twin primes form 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.
If it were the case that the sum diverged, then that fact would imply that there are infinitely many twin prime numbers. Because the sum of the reciprocals of the twin primes instead converges, it is not possible to conclude from this result that there are finitely many or infinitely many twin primes. Brun's constant could be an
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
only if there are infinitely many twin primes.
Numerical estimates
The series converges extremely slowly. Thomas Nicely remarks that after summing the first one billion (10
9) terms, the relative error is still more than 5%.
By calculating the twin primes up to 10
14 (and discovering the
Pentium FDIV bug
The Pentium FDIV bug is a hardware bug affecting the floating-point unit (FPU) of the early Intel Pentium processors. Because of the bug, the processor would return incorrect binary floating point results when dividing certain pairs of high- ...
along the way), 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 Patrick may refer to:
*Patrick (given name), list of people and fictional characters with this name
* Patrick (surname), list of people with this name
People
*Saint Patrick (c. 385–c. 461), Christian saint
* Gilla Pátraic (died 1084), Patrick ...
used all twin primes up to 1016 to give the estimate that ''B''2 ≈ 1.902160583104. Hence,
The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 1016. Dominic Klyve showed conditionally (in an unpublished thesis) that ''B''2 < 2.1754 (assuming the extended Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, wh ...
). It has been shown unconditionally that ''B''2 < 2.347.
There is also a Brun's constant for prime quadruplets. A prime quadruplet
In number theory, a prime quadruplet (sometimes called prime quadruple) is a set of four prime numbers of the form This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
Pri ...
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 ''B''4, is the sum of the reciprocals of all prime quadruplets:
:
with value:
:''B''4 = 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 primes, as prime pairs of the form (''p'', ''p'' + 4), which is also written as ''B''4. Wolf derived an estimate for the Brun-type sums ''B''n of 4/''n''.
]
Further results
Let be the twin prime constant. 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.
In popular culture
The digits of Brun's constant were used in a bid of $1,902,160,540 in the Nortel
Nortel Networks Corporation (Nortel), formerly Northern Telecom Limited, was a Canadian multinational telecommunications and data networking equipment manufacturer headquartered in Ottawa, Ontario, Canada. It was founded in Montreal, Quebec, ...
patent auction. The bid was posted by Google
Google LLC () is an American multinational technology company focusing on search engine technology, online advertising, cloud computing, computer software, quantum computing, e-commerce, artificial intelligence, and consumer electronics. I ...
and was one of three Google bids based on mathematical constants. Furthermore, academic research on the constant ultimately resulted in the Pentium FDIV bug
The Pentium FDIV bug is a hardware bug affecting the floating-point unit (FPU) of the early Intel Pentium processors. Because of the bug, the processor would return incorrect binary floating point results when dividing certain pairs of high- ...
becoming a notable public relations
Public relations (PR) is the practice of managing and disseminating information from an individual or an organization (such as a business, government agency, or a nonprofit organization) to the public in order to influence their perception. Pu ...
fiasco for Intel
Intel Corporation is an American multinational corporation and technology company headquartered in Santa Clara, California. It is the world's largest semiconductor chip manufacturer by revenue, and is one of the developers of the x86 seri ...
.
See also
* Divergence of the sum of the reciprocals of the primes
The sum of the reciprocals of all prime numbers diverges; that is:
\sum_\frac1p = \frac12 + \frac13 + \frac15 + \frac17 + \frac1 + \frac1 + \frac1 + \cdots = \infty
This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-centur ...
* Meissel–Mertens constant
The Meissel–Mertens constant (named after Ernst Meissel and Franz Mertens), also referred to as Mertens constant, Kronecker's constant, Hadamard– de la Vallée-Poussin constant or the prime reciprocal constant, is a mathematical constant in ...
Notes
References
*
*
*
* Reprinted Providence, RI: Amer. Math. Soc., 1990.
* Contains a more modern proof.
*
External links
*
*
* {{PlanetMath, urlname=BrunsConstant, title=Brun's constant
* 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
Sieve theory
Theorems about prime numbers