This is about understanding Congruence, I started with (Elementary Number Theory by David M. Burton), I am studying Chapter 4, Theory Of Congruence and the hard part is understanding the proof of Chinese Remainder Theorem and a Theorem related to solving system of linear congruences.
Further, please suggest If there is a better book for holding grip on solving problems related to Congruences and insightful reading on Number Theory somewhat written in Soviet-Mathematician-style, like no need to have a teacher.
Last, how should one approach Number Theory while learning it like Solving Problems, Understanding Proofs and Writing Proofs and Applying it in real life like Cryptography, programming problems.
Please help.
I read a lot of Joe Silverman's A Friendly Introduction to Number Theory. He actually says that he intended it for non-math majors. But don't let that deter you.
It's a fun book, well-written, thorough, covering lots of the standard number theory topics.
It's very accessible.
The section on the Chinese remainder theorem is well done. And it's noted that this implies Euler's phi function is multiplicative.
Let's see. Twin primes, Goldbach's conjecture, Fermat primes, Mersenne primes, Fibonacci numbers, big and little oh, cryptography, primality testing, Fermat and Euler's theorems, Carmichael numbers, congruence and modular arithmetic. Not in that order of course. The list goes on. Oh, and, big oversight on my part, the prime number theorem.
The last part covers Fermat's Last Theorem via elliptic curves and other trickery.
I have also heard good things about Baker's book.
And Davenport's The Higher Arithmetic sounds good.