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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

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Motivation often involves giving context for your results. Include some history of the problems that you study and indicate some of the relevant results that others have proven that relate to your results. Perhaps include some open problems as well as potential directions for future research.

Here are a couple of questions that a good motivation can answer: (1) Why are the problems you study of interest to a broad range of mathematicians? (2) What similar problems or similar results have others obtained that can be compared/contrasted with your results? (3) What makes your results or your approach novel?

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That is the point. Why are we interested in Mersenne primes and similar quantities? Marin Mersenne noticed that some positive integers that are one less than a power of two are prime. This is just a pattern. Mathematics is all about finding patterns! I feel like explaining the importance of Mersenne primes, amicable numbers, etc. is equivalent to explaining why we study numbers in general. It is like asking why do we study philosophy, read books, or even exist at all. – glebovg Oct 5 '12 at 23:19
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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
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... so in short, you should be able to give an explanation of why studying Mersenne primes is interesting beyond just "they are numbers and numbers are interesting." There are many results about Mersenne primes that cannot be proven about arbitrary numbers. These kind of results are what you want to highlight. What known facts about Mersenne primes are related to your work and your paper? What are the big open problems about Mersenne primes that you'd ultimately like to understand? (E.g. are there infinitely many Mersenne primes? What are the best algorithms to test their primality?) – Michael Joyce Oct 6 '12 at 0:35
Thank you for your advice! – glebovg Oct 6 '12 at 18:36

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