Difference between revisions of "Twin primes conjecture"
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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 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 | + | In 1915, [[Viggo Brun]] proved that the sum of reciprocals of the [[twin primes]] is convergent. This sum converges to [[Brun's constant]]. |
== Sources == | == Sources == | ||
*[http://mathworld.wolfram.com/TwinPrimeConjecture.html Math World] | *[http://mathworld.wolfram.com/TwinPrimeConjecture.html Math World] | ||
| − | + | [[Category:Number Theory]] | |
| − | [[Category: | ||
Latest revision as of 11:37, November 16, 2008
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Arenstorf published a purported proof of this conjecture, but a fatal defect was discovered in the proof and the paper was retracted.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 primes p such that p+2 is either a prime or the product of two primes (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 infinitely 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 to Brun's constant.