|
The twin prime conjecture is a famous problem in number
theory that involves prime numbers. It states:
There are an infinite number of primes p such that p + 2 is also prime.
Such a pair of prime numbers is called a twin prime. The conjecture has been
researched by many number theorists. Mathematicians believe the conjecture to be true, based only on numerical evidence and
heuristic reasoning involving the probabilistic distribution of primes.
In 1849 de
Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs
which have a distance of 2k. The case k = 1 is the twin prime conjecture.
Partial results
In 1940, Erdös showed that there is a
constant c < 1 and infinitely many primes p such that
p' − p < c ln p, where p' denotes the next prime
after p. This result was successively improved; in 1986 Maier showed that a
constant c < 0.25 can be used.
In 1966, Chen Jingrun showed that
there are infinitely many primes p such that p + 2 is a either a prime or a semiprime (i.e., the product of two primes). The approach he took involved a topic called sieve theory, and he managed to treat the
twin prime conjecture and Goldbach's conjecture in
similar manners.
Hardy-Littlewood conjecture
There is also a generalization of the twin prime conjecture, known as the Hardy-Littlewood conjecture (after
G. H. Hardy and John Littlewood), which is concerned with the distribution of twin primes, in analogy to the
prime number theorem. Let π2(x)
denote the number of primes p ≤ x such that p + 2 is also prime. Define the
twin prime constant C2 as
-
(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that
in the sense that the quotient of the two expressions tends
to 1 as x approaches infinity.
This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density
function of the prime distribution, an assumption suggested by the prime number theorem. The numerical evidence behind the
Hardy-Littlewood conjecture is quite impressive.
Serious problem found in potential proof
On May 26, 2004, Richard Arenstorf of Vanderbilt University
submitted a 38-page proof that there are, in fact, infinitely many twin primes. On June 3, Michel Balazard of University Bordeaux
reported that Lemma 8 on page 35 is false.[1] As is
typical in mathematical proofs, the defect may be correctable
or a substitute method may repair or replace the defect. Arenstorf withdrew his proof on June 8, noting "A serious error has been
found in the paper, specifically, Lemma 8 is incorrect".
See also
External Links
There Are
Infinitely Many Prime Twins - R. Arenstorf Withdrawn June 8, 2004.
|