Are there infinitely many twin primes like 11 and 13?

Question

Twin primes: ( 3 , 5 ), ( 5 , 7 ), ( 11 , 13 ), ( 17 , 19 ), ( 41 , 43 )... The Twin Prime Conjecture says there are infinitely many pairs of primes with difference 2. Recently (2013), Yitang Zhang proved theres a bound B = 70 , 000 , 000 such that infinitely many prime pairs differ by at most B. Then the Polymath project reduced this to 246. And Maynard showed theres infinitely many prime pairs with gap ≤ 600 without assuming the Elliott-Halberstam conjecture. But theres still a gap between 246 and 2. Is there any hope of reducing it to 2? Also: whats the current status of the conjecture? Is it "likely true" based on heuristics? I heard the Hardy-Littlewood conjecture gives a density estimate for twin primes.

Nour Hassan · Accepted Answer

Prime pairs such as 11 and 13 are called twin primes because they differ by 2. The big open question is: Are there infinitely many twin primes? Mathematicians strongly believe the answer is yes, but nobody has proved it yet. What has been proved is weaker but still important. Work by Yitang Zhang, James Maynard, Terence Tao, and the Polymath project showed that there are infinitely many prime pairs whose gap is bounded by some fixed number. In simple terms, primes keep appearing close together infinitely often, but we have not forced the gap all the way down to 2. So: - infinitely many prime pairs with some bounded gap: proved, - infinitely many twin primes with gap exactly 2: still open. That is why twin primes are famous: the pattern is easy to see, but very hard to prove forever.

Are there infinitely many twin primes like 11 and 13?

1 Answer

Are there infinitely many twin primes like 11 and 13?

1 Answer