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New page: The '''twin primes conjecture''' states that there are infinitely many twin primes. It is one of the great unsolved problems in mathematics, as it has never been proven. In 2004,...

The '''twin primes conjecture''' states that there are infinitely many [[twin primes]]. It is one of the great unsolved problems in [[mathematics]], as it has never been proven.

In 2004, Professor Arenstorf published a purported proof of this conjecture, but a fatal defect was discovered in the proof and the paper was retracted.

In 1966, Chen Jingrun showed that there are infinitely many [[prime]]s p such that p+2 is either a [[prime]] or the product of two [[prime]]s (a "[[semiprime]]"), relying on a sieve theory. This resulted in defining a "Chen prime" to be a prime p such that p+2 is either a prime or a [[semiprime]]. In 2005, Ben Green and Terence Tao proved that there are [[infinite]]ly many three-term arithmetic progressions of Chen primes.

In 1915, [[Viggo Brun]] proved that the sum of reciprocals of the [[twin primes]] is convergent. This sum converges on [[Brun's constant]].

== Sources ==

*[http://mathworld.wolfram.com/TwinPrimeConjecture.html Math World]

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