There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. 19). It is not known if there
are an infinite number of such primes
(Wells 1986, p. 41; Shanks 1993, p. 30), but it seems almost certain to
be true. While Hardy and Wright (1979, p. 5) note that "the evidence, when
examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219)
states even more strongly, "the evidence is overwhelming," Hardy and Wright
also note that the proof or disproof of conjectures of this type "is at present
beyond the resources of mathematics."
Arenstorf (2004) published a purported proof of the conjecture (Weisstein 2004). Unfortunately, a serious error was found in the proof. As a result, the paper was retracted and the twin prime conjecture remains fully open.
The conjecture that there are infinitely many Sophie Germain primes, i.e., primes such that is also prime, is very closely related (Shanks
1993, p. 30).
A second twin prime conjecture states that adding a correction proportional to to a computation of Brun's constant ending with
will give an estimate with error less than .
An extended form of this conjecture, sometimes called the strong twin prime conjecture
(Shanks 1993, p. 30) or first Hardy-Littlewood
conjecture, states that the number of twin
primes less than or equal to is asymptotically
equal to
where is the so-called twin primes constant (Hardy and Littlewood 1922). The value
of is plotted above for
, with indicated in
blue and taking .
This conjecture is a special case of the more general k-tuple conjecture (also known as the first Hardy-Littlewood conjecture), which corresponds to the set
.
Arenstorf, R. F. "There Are Infinitely Many Prime Twins." 26 May 2004.
http://arxiv.org/abs/math.NT/0405509.
Guy, R. K. "Gaps between Primes. Twin Primes." §A8 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag,
pp. 19-23, 1994.
Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44,
1-70, 1923.
Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford,
England: Clarendon Press, 1979.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton
University Press, pp. 30-31, 2003.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag,
pp. 261-265, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed.
New York: Chelsea, p. 30, 1993.
Tenenbaum, G. "Re: Arenstorf's paper on the Twin Prime Conjecture." 8 Jun 2004. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0406&L=nmbrthry&F=&S=&P=1119.
Weisstein, E. W. "Twin Prime Proof Proffered." MathWorld headline news, June 9, 2004. http://mathworld.wolfram.com/news/2004-06-09/twinprimes/.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers.
Middlesex, England: Penguin Books, p. 41, 1986.
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