What is the importance of Polignac’s conjecture? 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?
 A: Speaking from the point of view of someone interested in prime number theory, the fact that the statement "there are infinitely many pairs of primes $p,q$ with $|p-q| \leq 6$" is provable conditionally (on the extended Elliott-Halberstam conjecture) but the twin-prime conjecture is NOT provable even if we throw everything we have at it, both proven and conditional, is quite remarkable. The key is that prime gaps of size at most $6$ do not require us to break the parity barrier, while twin primes does.
A: As far as I know, the twin primes conjecture doesn't have applications.
It is considered interesting because it is an extreme example of the kind of simple to state, hard to solve problems that are favored in number theory. It's also one of the older open problems in number theory, though surely not the very oldest. Finally, it's considered a good test of the strength of our methods.
Any approach that solves the twin primes conjecture will likely have applications to other problems on the distribution of prime numbers. In particular, it's reasonable to guess that it would lead to a solution to Goldbach's conjecture, at least for sufficiently large n. It may involve progress on Chowla's conjecture and the randomness of the Mobius function. It could also lead to progress on pair correlation of the zeroes of the Riemann zeta function.
