The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes).
We know that as numbers get larger, primes become less frequent and twin primes rarer still.
In 1919, Viggo Brun showed that the sum of the reciprocals of the twin primes converges to a sum, known as Brun’s constant (approximately 1.90216054).
In 2003, the next big breakthrough occurred. Daniel Goldston and Cem Yıldırım published a paper, “Small Gaps Between Primes,” that established the existence of an infinite number of prime pairs within a small difference. Yitang Zhang showed in 2013 (Bounded gaps between primes) that, without any assumptions, there were an infinite number of pairs of primes differing by at most 70 million (this bound was improved to 246 in 2014).
Even though it is clear what this conjecture asserts, I cannot grasp yet why it is so relevant as to be considered one of the biggest open problems in number theory. Which are its implications? What are the problems that will be solved with the same possible solutions?