# Subject advice in Number Theory [closed]

At my University, we have the optional feature to write a project like a Bachelor Thesis. This semester have finished and I would like to work in the summer in project like this. So, I'm searching for a topic and I would appriciate your opinion.

More specifically, I would like to work on a topic relative with Number Theory (like Diophantine Equations etc) or secondarily with Abstract Algebra.

Question. Could you suggest me some topics accompanied by the corresponding bibliography?

PS 1: I should inform you that I have been taught Elementary Number Theory, Abstract Algebra (Group & Ring Theory) and some Galois Theory (all in an undergraduate level, please let me know if you want more information about this).

PS 2: I hope this post is not off topic. Otherwise inform me and I will delete it.

• You might get some ideas by browsing (open) problem collections, e.g. the books: D. Shanks, Solved and unsolved problems in number theory, and R. K. Guy, Unsolved problems in number theory. – Bill Dubuque Jun 30 '18 at 18:34
• @Number Thank you for your comment. Ok, I ll browse it, but I would preffer to study something in a more bibliographic than a reasearch way. – Chris Jun 30 '18 at 18:36
• Yeah, I think that this is off-topic for the site. Most questions that seek personal advice are. This is a good question for your academic adviser though, or for a faculty member at your university, since they presumably know you better than the people on this site, and you can sit down and have a conversation about what to study. And when talking with a faculty member, if they suggest something they know a bit about, they may be willing to continue advising you as you read on the topic. – Mike Pierce Jun 30 '18 at 19:49
• Nah, the fact that it's off topic is more FYI for your future posts (Welcome! 😊) Leave your question for now. Let anyone else chime in who wants to, and if other users of the site agree with me then this question will eventually be closed and deleted. – Mike Pierce Jun 30 '18 at 20:59
• Is this Bachelor Thesis supposed to be original research (like proving new results or investigating open problems)? Or is it a summary/survey of an interesting problem/field of math? I had to do a Senior Thesis my last semester of college and wrote about Continued Fractions. Very interesting field of number theory with many far reaching applications and interesting results (including one from Mr. Galois himself at the age of 18). – ruferd Jul 2 '18 at 17:24

As mentioned in the comments, I used the topic of Continued Fractions when I had a similar paper to write. The applications include: rational approximations to irrational numbers, Diophantine Equations, Pell's Equation, Factoring Algorithms, etc.

Some of my sources included:

William Stein: "Elementary Number Theory: Primes, Congruences, and Secrets"

A. Rockett and P. Szusz: "Continued Fractions"

Hugh L. Montgomery, Ivan Niven, and Herbert S. Zuckerman: "An Introduction to the Theory of Numbers"

• Thank you very much for all the useful information! – Chris Jul 3 '18 at 12:35

I think it would be a good Bachelor topic to explain, at a high-level, the MRDP theorem (Matiyasevich–Robinson–Davis–Putnam), which settled Hilbert's 10th problem:

A set of integers is Diophantine if and only if it is computably enumerable.

And its many consequences and refinements. E.g., just $11$ integer variables are enough to lead to undecidable Diophantine equations.1

And there are many open problems2 to which you could refer. For example, find all integer solutions to $$\binom{x}{2} = \binom{y}{5} \;.$$

(Image from AMS Notices: Julia Robinson and Hilbert’s Tenth Problem.)

1 James Jones. "Undecidable diophantine equations." Bull. Amer. Math. Soc. (N.S.). Volume 3, Number 2 (1980), 859-862.

• Thank you very much for your answer! I ll have a look! – Chris Jun 30 '18 at 23:50