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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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?