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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.

Shantanu Sharma
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  • There are many good book recommendations for self learning number theory on this site, e.g., here. The best method is to do examples and counterexamples. Then you can make every definition and every statement explicit, until you understand it. – Dietrich Burde May 05 '20 at 11:38
  • @DietrichBurde need something more subtle readings. – Shantanu Sharma May 05 '20 at 12:00
  • @DietrichBurde Already mentioned that something in soviet style. I know Mathstackexcahnge is great. Those are good recommendations which you shared but I am looking for something different. – Shantanu Sharma May 05 '20 at 12:16
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    Are you at (or near) a university? Is there a faculty member you could discuss this with? you'll get better advice from someone you can sit down and talk to, than you will from random people on a website. – Gerry Myerson May 05 '20 at 12:36
  • You need to be more specific about precisely what you have difficulty understanding in order for us to best help you. Section 4.4 contains a common (unmotivated) proof of CRT. See here for a motivated conceptual derivation of that CRT formula. The section ends with a theorem showing how to solve a system of two equations in two variables $\bmod n$ by elimination (or, equivalently, by Cramer's rule, e.g. see here and here). – Bill Dubuque May 05 '20 at 16:22
  • What is Soviet-mathematician style? – mpnm May 08 '20 at 05:17
  • Thanks @DietrichBurde I underestimated Kenneth Rosen's Elementary Number Theory And Its Applications. Its a awesome book for Number Theoory. Thank you. – Shantanu Sharma May 31 '20 at 07:16
  • @Gone Thanks ,that helped me to get CRT proof much better. – Shantanu Sharma May 31 '20 at 07:18

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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.