Intereting Posts

Books about the Riemann Hypothesis
Existance and uniqueness of solution for a point with fixed distances to three other points
Leibniz rule, multiple integrals
Hopf's theorem on CMC surfaces
Is there a name for the “famous” inequality $1+x \leq e^x$?
Trouble with gradient intuition
Clarification regarding white space in rules of inference
$\int_0^{a} x^\frac{1}{n}dx$ without antiderivative for $n>0$
How many different basis' exist for an n-dimensional vector space in mod 2?
Determine all real polynomials with $P(0)=0$ and $P(x^2+1)=(P(x))^2+1$
A generalization of the connected sum of links
Question regarding integral of the form $\oint\limits _{s_{R}}e^{i\cdot k\cdot z}f\left(z\right)dz$.
Proof about a Topological space being arc connect
Big List of Erdős' elementary proofs
There is no smallest rational number greater than 2

Since summer comes with a lot of spare time, I’ve decided to select a mathematical subject I want to learn as much as possible about over the next three months. That being said, number theory really caught my eye, but I have no prior training in it.

I’ve decided to conduct my studying effort in a library; I prefer real books to virtual ones, but as I’m not allowed to browse through the books on my own, I have to know beforehand what I’m looking for and this is where I’m kind of lost. I’m not really sure where to start.

Basically I wanna know about the following:

- HCF/LCM problem
- Prove that if the square of a number $m$ is a multiple of 3, then the number $m$ is also a multiple of 3.
- How to show $\binom{2p}{p} \equiv 2\pmod p$?
- congruence issue
- Linear diophantine equation $100x - 23y = -19$
- $\sqrt{13a^2+b^2}$ and $\sqrt{a^2+13b^2}$ cannot be simultaneously rational

What are the prerequisites?

(I’m currently trained in Linear Algebra, Calculus, Complex Analysis – all on an undergraduate level )

Can you recommend some reading materials?

Thank you.

- How, if at all, does pure mathematics benefit from $2^{74207281}-1$ being prime?
- Find the value of $\left$
- Sieving integers
- Proof of irrationality of square roots without the fundamental theorem of arithmetic
- Prove or disprove: For every integer a, if a is not congruent to 0 (mod 3), the a^2 is congruent to 1 (mod 3)
- Intuition and Stumbling blocks in proving the finiteness of WC group
- Show $(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdots)^2 = 1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49} + \cdots$
- unique factorization of matrices
- When is $\sin x$ an algebraic number and when is it non-algebraic?
- Modulo of a negative number

I will do exactly the same thing. I just finished my degree in mathematics but in our department there is not a single course of Number Theory, and since I will start my graduate courses in October I thought it will be a great idea to study Number Theory on my own. So, I asked one of my professors, who is interested in Algebraic Geometry and Number Theory, what would be a textbook that has everything an undergraduate should know about Number Theory before moving on. He told me that A Classical Introduction to Modern Number Theory by *Kenneth F. Ireland and Michael Rosen* is the perfect choice. He also mentioned that I should definitely study chapters 1-8,10-13 and 17. Another book that he mentioned was A Friendly Introduction to Number Theory by *Joseph H. Silverman*. He emphasized though that this book is clearly an introduction whereas the previous one gives you all the tools you need in order to study many things that are connected to Number Theory. I hope that this helped you!

In my opinion Hardy &Wright’s book on Number Theory is not the best possible book for someone “who has no prior training in Number Theory”, I would suggest the following books.

Elementary Number theoryby David M. Burton.

Number Theory A Historical Approachby John H. Watkins

Higher Arithmeticby H. Davenport

All the books are well-written. I think that if you are a beginner, and if you are interested in the historical aspects of Number Theory as well, you may first look at the second book. Although Burton’s books also have some historical background in each chapter. I would suggest reading Davenport’s books a bit later when you have a fair grasp of the subject.

Also I suggest you to look at the suggestions given at this post.

One of the best is *An Introduction to the Theory of Numbers* by Niven, Zuckerman, and Montgomery.

I would recommend An Introduction to the Theory of Numbers By G.H. Hardy and E.M. Wright .

Dover publishes many number theory titles. At \$10-\$15 each they’re a bargain – no need even to look for the Amazon discount. You can get several and jump back and forth among them to get different perspectives on each topic. You can write yourself notes in the margins. Take them to the library to read.

This is a standard old undergraduate text:

Elementary Number Theory: Second Edition

(Underwood Dudley)

http://store.doverpublications.com/048646931x.html

Way off the beaten path, but fun, with a rarely encountered (in elementary texts) proof of quadratic reciprocity:

An Adventurer’s Guide to Number Theory

(Richard Friedberg)

http://store.doverpublications.com/0486281337.html

I can certainly recommend *Elementary Number Theory* by Gareth A Jones et al. It will get you started and then you can move onto more advanced texts. It’s a very short book (about 300 pages) which means you can easily read through the whole text-a very good choice for self study. As for pre-requisites, a good grasp of algebra will probably do.

- First Order Language for vector spaces over fields
- What does a “half derivative” mean?
- Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$.
- How to prove $\mathbb{Z}=\{a+b\sqrt{2}i\mid a,b\in\mathbb{Z}\}$ is a principal ideal domain?
- eigenvalues of a matrix $A$ plus $cI$ for some constant $c$
- Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$
- Integral $\int_0^{2\pi}\frac{dx}{2+\cos{x}}$
- Characterization of Dirac Masses on $C(,\mathbb{R}^d)$
- A calculation that goes awfully wrong if we let $\pi=22/7$
- Showing that $n$ is pseudoprime to the base $a$
- Closed form of infinite product $ \prod\limits_{k=0}^\infty \left(1+\frac{1}{2^{2^k}}\right)$
- Relationship between Legendre polynomials and Legendre functions of the second kind
- If $\int_{0}^{x}f(t)dt\rightarrow \infty$ as $|x|\rightarrow \infty\;,$ Then every line $y=mx$ Intersect $ y^2+\int_{0}^{x}f(t)dt=2$
- What is the value of lim$_{n\to \infty} a_n$ if $\lim_{n \to \infty}\left|a_n+ 3\left(\frac{n-2}{n}\right)^n \right|^{\frac{1}{n}}=\frac{3}{5}$?
- Find all Laurent series of the form…