How applicable is Goldbach's conjecture to real world scenarios?

Question

In what scenarios is Goldbach's conjecture, that all even numbers greater than 4 are the sum of two prime numbers; a natural conjecture to research?

Answer 1

In Mallory's immortal words: Because it's there.

After the proof of Fermat's last theorem, Goldbach's conjecture is probably the leading example of a conjecture which is extremely simple to explain -- even a grade schooler will be able to understand what the question is -- but where no answer has been produced even after centuries of concerted effort by a host of very smart people.

For some people, that very property makes it impossible not to think about the problem and try to figure out ways to attack it. There is something infuriating about such a simple statement resisting all the human ingenuity we can throw at it.

A different question is why universities would pay people to think about this particular problem. Here one can point either to the fact that (as Patrick notes) the mathematics that's created while attacking hard problems such as this often turns out to have broader applications -- or to the fact that giving academics the freedom (at least collectively) to work on whatever problem catches their fancy has historically been very successful in getting useful and interesting breakthroughs that nobody could have anticipated.

Even if Goldbach's conjecture in particular could be reliably know to lead nowhere useful, we don't have any workable general way to force basic research in a useful direction that works better than "follow your curiosity". Trying to stomp out individual lines of inquiry because they're deemed fruitless would generally create too many secondary problems (general miscontent, researchers who don't really care about Goldbach wondering whether they will be next, and so forth) to be worth it.

Answer 2

The point of such conjectures are often not the result themselves ; more often than not the knowledge of the veracity of such statements is completely uninteresting. The interest lies in the methods of proof to argue against such statements ; they often lead to developing new theories and inspire mathematicians to create new kinds of arguments which can then apply to other situations where they may hopefully be useful. Think of it as a training ground for mathematicians.

Hope that helps,