Good theory in number theory for deep understanding

In summary, the conversation discusses the search for a good book on number theory that provides a deep understanding of concepts, similar to Kiselev in geometry and Apostol in calculus. The book "Introduction to Theory of Numbers" by I. Niven and H. S. Zuckerman is mentioned, but it is not suitable for the individual's purpose as it lacks explanations and encouragement for problem-solving. The individual is looking for a book with good theory and asks for recommendations. Apostol's Number Theory book and the follow-up on modular forms and Dirichlet series are suggested as masterpieces. Other recommended books include Ireland and Rosen's classic text and Davenport's Multiplicative Number theory.
  • #1
simpy
18
0
Does anybody of a good book in number theory for deep understanding of concepts like kiselev in geometry and apostol in calculus ? I have 'introduction to theory of numbers' by I. Niven and H. S. Zuckerman but I feel it is not suitable for my purpose , there is no description of 'why we did this' in the book and in examples too the book does not encourage the reader to solve the problem on his/her own.
All I mean by above is that although the questions' quality is good in the book but theory is not and I need a good theory book.
So, if you know any book that can meet my purpose, share it here.
 
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  • #2
Apostol's Number Theory book is a masterpiece ..
The follow-up on Modular forms and Dirichlet series .. is also a masterpiece.
Wakabaloola
 
  • #3
Well there's the classic text by Ireland and Rosen, I think this book alongside Davenport's Multiplicative Number theory is a good start in theory respect.
 

1. What is the importance of good theory in number theory for deep understanding?

Good theory in number theory is crucial for deep understanding because it provides a framework for organizing and analyzing the vast amount of information in this complex mathematical field. It allows for the development of new concepts and techniques, and helps to uncover connections between seemingly unrelated topics.

2. How do mathematicians determine if a theory in number theory is considered "good"?

A good theory in number theory is typically characterized by its ability to explain and predict mathematical phenomena, its elegance and simplicity, and its potential for further development and application. Additionally, it should be consistent with existing theories and supported by empirical evidence.

3. Are there any specific techniques that are commonly used in the development of good theories in number theory?

Yes, there are several common techniques used in the development of good theories in number theory, including mathematical induction, proof by contradiction, and the use of modular arithmetic. These techniques help to establish the validity and rigor of the theory.

4. Can good theory in number theory be applied to practical problems?

Yes, good theory in number theory has numerous practical applications in fields such as cryptography, computer science, and physics. For example, number theory is essential in developing secure encryption algorithms for online transactions and data protection.

5. How can a deep understanding of number theory benefit society?

A deep understanding of number theory can lead to advancements in various fields, such as technology, finance, and medicine. It also helps to improve problem-solving skills and critical thinking, making it an important area of study for developing future leaders and innovators.

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