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MATHEMATICS.
MATHEMATICS
CONCEPTS OF TWIN PRIMES.
MATHEMATICS.
Contents
CONCEPTS OF TWIN PRIMES......................................................................................................................... 3
Introduction. ............................................................................................................................................. 3
Brun’s Theorem......................................................................................................................................... 3
Conjectures. .............................................................................................................................................. 3
First Hardy-Littlewoods Conjecture. ..................................................................................................... 3
Polignac’s conjuncture. ......................................................................................................................... 3
Large Twin primes. .................................................................................................................................... 3
Other Elementary properties. ................................................................................................................... 3
Conclusions. .............................................................................................................................................. 4
Bibliography .................................................................................................................................................. 4
MATHEMATICS.
CONCEPTS OF TWIN PRIMES.
Introduction.
Brun’s Theorem.
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous
result, called Brun’s theorem, was the first use of the Brun sieve and helped initiate the development of
modern sieve theory. The modern version of Brun’s argument can be used to show that the number of
twin primes less than N does not exceed.
For some absolute constant C > 0. In fact, it is bounded above by,
C1 N
(1+0) Log Log N
(Log N)2
Log N
Where C 1 = 8 C 2 is the twin prime constant, given below.
Conjectures.
First Hardy-Littlewoods Conjecture.
Polignac’s conjuncture.
Large Twin primes.
Other Elementary properties.
MATHEMATICS.
Conclusions.
Bibliography
Goldston, D. A., Motohashi, Y., Pintz, J., & Yıldırım, C. Y. (2006). Small gaps between primes exist. Japan:
Series A.
Heini Halberstam, a. H.-E. (2010). Sieve Methods. Dover Publications.
McKee, M. (14 May 2013). "First proof that infinitely many prime numbers come in pairs.
Tao, T. ( June 4, 2013). Polymath proposal: bounded gaps between primes.
Zhang, Y. (2014). Annals of Mathematics-Bounded gaps between primes.