Big advance on simple-sounding math problem was a century in the making

How do mathematicians come to focus on seemingly arbitrary quesrions:

> another asks whether there are infinitely many pairs of primes that differ by only 2, such as 11 and 13

Is it that many questions were successfully dis/proved and so were left with some that seem arbitrary? Or is there something special about the particular questions mathematicians focus on that a layperson has no hope of appreciating?

My best guess is questions like the one above may not have any immediate utility, but could at any time (for hundreds of years) become vital to solving some important problem that generates huge value through its applications.

There's something quite interesting about the problems in number theory especially. The questions/relationships sometimes don't seem useful at all and are later proven to be incredibly useful. Number Theory is the prime example of this. I believe there's a G H Hardy quote somewhere, about Number Theory being obviously useless, but could only find it from one secondary source, although it does track with his views expressed in A Mathematician's Apology[1] - "The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics."

You can find relationships between ideas or topics that are seemingly unrelated, for instance, even perfect numbers and Mersenne primes have a 1:1 mapping and therefore they're logically equivalent and a proof that either set is either infinite or finite is sufficient to prove the other's relationship with infinity. There's little to no intuitive relationship between these ideas, but the fact that they're linked is somewhat humbling - a fun quirk in the fabric of the universe, if you will.

[1] https://en.wikipedia.org/wiki/A_Mathematician's_Apology

> No one has yet found any war-like purpose to be served by the theory of numbers or relativity or quantum mechanics, and it seems very unlikely that anybody will do so for many years.

G.H.Hard. Eureka, issue 3, Jan 1940

Less than 100 years later, we stand waiting for nuclear bombs guided by GPS to be launched when the authorization cryptographic certificate is verified.

Nuclear weapons (based on quantum mechanics and special relativity) were used less than 6 years after that quote.