I am currently writing a couple of undergraduate papers about primes and irrational numbers, and my advisor keeps saying that I need to motivate the topics and include a discussion at the end. Can anyone explain how to motivate Mersenne primes and/or a new irrationality criterion. How do mathematicians motivate theorems about numbers such as Mersenne primes? Apparently, I need to explain why my results are useful, but is it not obvious? Why else do we prove theorems?

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This is not about Mersenne primes, but about the sort of related Amicable Numbers. Someone writing about $1000$ years ago said that he had tested the erotic power of amicable numbers by feeding $220$ to his lover while eating $284$ himself. An early application of number theory. – André Nicolas Oct 5 '12 at 22:40

You should take for granted that your reader understands the importance of studying prime numbers and how difficult they are to characterize. There are techniques and algorithms available to detect whether $2^p-1$ is prime, which have been used to find Mersenne primes that are at the given time the largest known prime number. Finding large prime numbers is exciting because it is so challenging to do. It tests the limits of our ability to detect primality, as well as underscore just how little we know about this... (cont.) – Michael Joyce Oct 6 '12 at 0:32