Why it is important to find largest prime numbers? It always takes a lot of effort and money to find the next largest prime number. Why is it so important to do this work and what is the application those numbers?
 A: Reason number 1: scientific curiosity. 
Reason number 2 (the economic reason): The RSA encryption algorithm requires large prime numbers to make secure data transmissions.  The larger the primes, the more secure the transmission.
A: Just to add to the previous answers: Usually, part of the discovery of these mathematical curiosities is not the result itself, but the new or improved method for finding these new results. It's not just about finding the number which is meaningless in itself, but it's about showing that mathematical techniques have advanced so much that we can even show that these incredibly huge numbers are prime.
Similarly, who cares that we planted a flag on the moon, or that we sent people there at all? It's all about showing that we can do this.
A: Because they're there.
Euclid called our attention to the fact after "because" in the third century BC, and Hillary called our attention to the appropriateness of "because" in the 20th century AD.
Also, large primes play a role in cryptography, which protects your bank P.I.N. number from thieves.
A: Finding Primes is basically a giant benchmark and test platform for large computers and distributed computing systems. It has several advantages as a benchmarking and experimental problem


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*Its a simple and well understood problem (Is this number prime? No, ok next one)

*There are many potential algorithms that can be directly compared

*Its simple enough that it can be run on any computational machine, providing a standard benchmark for performance


Think of it as a "test-system" for computer scientists. By learning how to efficiently distribute work over millions of computers to find something like the next largest prime, we can research new techniques for doing so more efficiently. Imagine a more complex problem, like weather prediction, here we not only need to figure out a good algorithm to solve such a complex system, but also how to distribute it efficiently. Through research on primes and other simple (as in, well defined) computational problems, the first part is basically taken care of, we can learn to more efficiently corral  computers to solve large problems. Because its so easy to objectively analyze the difference in efficiency (literally, how long it takes), we can compare the relative merits of different distribution schemes, both in software (multithreading) and in hardware (different memory/system architectures). 
As for research into prime algorithms themselves, being able to find large primes is needed for most canonical encryption schemes, larger primes are harder to factor and therefore more secure. Its also a research field in number theory.
As for specific projects dedicated to finding primes, I'd wager that its just out of general curiosity and competition. 
A: Re the original question: I'd hazard a guess that, for almost the entire world, the question 'why' simply wouldn't arise, since the work makes zero contribution to their welfare and the future of humanity. It's probably safe to assume that it has subjective value for the few who engage themselves in the task.
A: This question presumes that finding large prime numbers is important, which is of course dead wrong. After all mathematics is all about proving theorems, simply knowing that some large number is prime is of no value whatsoever. Doesn't tell you anything. And apart from this, the widely shared opinion is that this is second-rate work, as it merely depends on pushing some button.
