r/learnmath • u/RectallyDisabled New User • 1d ago
TOPIC What does it take to study number theory thoroughly?
I'm 16, in my last year of high school. Over the last few years, I've been working on mathematics and physics on my own because I love them so much. Since I'm nearly graduating from high school, I was wondering what's really holding me back from doing research. I'm on track to finish Calc 1, 2, and 3 by the time I graduate (late 17s), but what comes after that? I'd like to study just about as much as I can from different branches, but once I'm decent in a variety of them, I'll research number theory. My question may not be very straight with how I'm wording this post, but I'm only really wondering what material realistically stops me from studying number theory at an 'advanced' level (as for the other branches, I'll pick up some books and go through them slowly). Any resources are helpful, assuming a basis of Calc 1 + fraction of Calc 2, basic number theory such as modular arithmetic, Fermat's little theorem, Euler's theorem, and other (light) theorems.
> When I said advanced: 'Advanced' is subjective, but to me, that's being able to write a worthy master's thesis in the field.
And of course I expect this to be a years-long challenge (an enjoyable one, at that) that I'm willing to take on
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u/Chocadooby New User 1d ago edited 23h ago
To really understand elementary number theory in a universal way it helps to know a bit of group and ring theory. You also need to know a bit of number theory and congruences to understand which elements of a cyclic group are generators of the group. But that is putting the cart before the horse. Just get "A Friendly Introduction to Number Theory" by Silverman.
Alternatively you can read a book on proofs like "Book of Proof" (free online!) by Richard Hammack and then tackle an introductory book on abstract algebra like "Abstract Algebra" by Herstein or "Abstract Algebra, A First Course" by Saracino. A basic introduction to abstract algebra will contain the fundamental intuitions of elementary number theory in a more general context. The typical math curriculum takes you through calculus, then a course on linear algebra, then an intro to proofs course, and then an introductory abstract algebra course. But if you have a good foundation in (highschool) algebra and problem solving, you can just jump straight into a a book on proof writing and then from there go into abstract algebra.
Some people can jump into proof-based math without explicitly studying how proofs work, but the American curriculum is very mechanical and procedural, and studying proofs lubricates what would otherwise be a rough entry.
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u/Low_Breadfruit6744 Bored 1d ago
Learn some complex number calculus. Learn how the riemann hypothesis is linked to prime numbers
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u/Legitimate_Log_3452 New User 1d ago
In my opinion, I think you should spend your time focusing on other things before number theory. Number theory is heavily reliant on other fields of math, so if you want to go deep into it, you’re going to need to anyway. For example, number theory generally uses abstract algebra, and analytic number theory uses analysis/complex analysis.
I dunno though. Take this with a grain of salt. I do analysis (like methed up calculus), not number theory
How are your proofs? I’d work on those first. You might be able to find an intro to number theory + intro to proofs book.
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u/OkHand7497 New User 1d ago
The idea that you might learn enough number theory to be able to write or understand a master's thesis in a year's time is rather implausible. I'd suggest asking for a more realistic reference for a specific area of number theory you're interested in.
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u/RectallyDisabled New User 1d ago
Years in plural of course. As for finishing calculus, I don't think finishing it in 4 months from now is unrealistic. My question was vague and so I wasnt sure if I should bother asking it, but a rough idea of what tools are required in modern analytical number theory is what I am looking for, and what resources can teach me, considering my basis.
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u/OkHand7497 New User 1d ago
Analytic number theory would necessitate some complex analysis I'm fairly certain - I think proofs of the prime number theorem rely on it. I'm not a number theory specialist myself. Dirichlet's theorem takes a reasonable deal of analysis too.
You may well be able to get your head around Euler's proof that
1/2 + 1/3 + 1/5 + 1/7 + ... + 1/p + ...
diverges without too much work. Perhaps something to aim for so as to put your toe in these waters?
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u/InfernicBoss New User 10h ago
people spend 5+ years studying number theory to write a thesis on it, so i would say u should set ur sights on basic undergrad math first before researching advanced math
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u/TheRedditObserver0 Grad student 23h ago
You will mostly need complex analysis and abstract algebra, with a greater focus on one or the other depending on whether you want to study analytic or algebraic number theory.
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u/localizeatp New User 1d ago
It takes patience and practice.
A large portion of number theory doesn't even require calculus. Try out Ireland and Rosen and let me know if it's too much.