## Twin-Primes Conjecture Now Even More Likely True

Posted on: June 10th, 2013 by Aaron Lauve

Three cheers for the indefatigable (! ! !). Many stars have taken their turn at playing the un-heralded protagonist, David, the Tortoise, Einstein. Add another one in Dr. Yitang Zhang.

Conjecture (de Polignac, 1849): There are infinitely many natural numbers n so that (n–1) & (n+1) are both prime.

You will be hard pressed to find any mathematician anybody who things the twin-primes conjecture is false, and examples abound, e.g.,  5 & 7,  17 & 19,  or  2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 + 1, but very little progress(*) has ever been made on actually proving the conjecture….

See also Nature‘s article about Zhang, from which the image was taken.

(*)  I should say little positive progress has been made. (a) In (Brunn, 1915), we learn that the sum of the reciprocals of twin primes converges. (b) From the prime number theorem (Legendre, 1797, de la Vallée-Poussin, 1896), we know that the average gap between primes grows like n/(ln n). So there aren’t too many twin primes, and they should be, on average, harder and harder to find.

(**) Anticlimactic short answer: the twins are now at most 70 million integers apart.