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Im an undergraduate in the mathematics field ..So i wanna be alittle more productive and wanted to do an essay or project mostly on number theory or Algebra(Rings or Groups) and i want to ask if you have to suggest some good topics for undergraduate level..and give me some tips on how to start.(and references)

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@Apurv Why remove the relevant tags? –  user1729 Feb 27 at 12:05
    
@user1729, the question is not on number theory, it is related to personal opinion and career development. So I removed the irrelevant tags. –  Apurv Feb 28 at 4:14

4 Answers 4

up vote 6 down vote accepted

One possibility is to study the so called "Euclidean proofs" for Dirichlet's theorem on arithmetic progressions, which are sometimes possible and are much easier than Dirichlet's methods form analytic number theory. However, it is an interesting question to say what an "Euclidean proof" should be, and for which arithmetic progressions $an+b$ it works. For references, see for example Ram Murty, M.; Thain, N.: Prime Numbers in certain Arithmetic Progressions.

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Oh! Thank you! It is interesting! –  Manolis Lyviakis Feb 27 at 10:59

You may try to understand the key points in the AKS primality test.

The book Primality Testing in Polynomial Time by Dietzfelbinger is very nice.

There is also the more recent, and probably simpler, Primality Testing for Beginners by Rempe-Gillen and Waldecker.

See also the survey It is easy to determine whether a given integer is prime by Granville.

A related book is The Joy of Factoring by Wagstaff.

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It's a bit of a challenge but perhaps you could summarize the activity following Yitang Zhang's recent paper on the gaps between primes. A friend of mine is doing so as his Master's dissertation; it's doable and it'd get you up-to-date. You'd have to black-box a few results as an undergraduate though.

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Ohh yes it was an awsome result! –  Manolis Lyviakis Feb 27 at 11:02

A wonderful book is Proofs from THE BOOK by Ziegler and Aigner (the capitals are in the title - I am not actually shouting). This book contains lots of wonderfully elegant proofs from number theory, geometry, analysis and so on. The proofs are almost always at a low-enough level for a good undergraduate student to understand (the idea is that they are simple and elegant, and therefore are understandable!). Clicking on the following link will download a table of contents for the 4th edition: link.

The name Proofs from THE BOOK is in honour of Paul Erdős. From wikipedia: "(The book is) a book in which God had written down the best and most elegant proofs for mathematical theorems. Lecturing in 1985 he said, You don't have to believe in God, but you should believe in The Book...He accused (God) of hiding his socks and Hungarian passports, and of keeping the most elegant mathematical proofs to himself. When he saw a particularly beautiful mathematical proof he would exclaim, This one's from The Book!."

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