- #1

matqkks

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I am looking for applications which will motivate the student in this subject.

Are there good resources on elementary number theory?

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- Thread starter matqkks
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In summary, the conversation discusses the topics that should be included in a first course in elementary number theory, as well as potential applications and resources. Some suggested topics include rational approximations, squaring the circle, and public-key cryptography, with a recommendation to also cover modular arithmetic. The conversation also mentions a book by Graham Everest and Thomas Ward as a potential resource for ideas.

- #1

matqkks

- 285

- 5

I am looking for applications which will motivate the student in this subject.

Are there good resources on elementary number theory?

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- #2

Fernando Revilla

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An Introduction to Number Theory

- #3

Evgeny.Makarov

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Another interesting topic that touches number theory is squaring the circle. It took us a third of a semester during the fourth year in college to cover the theorem that $\pi$ is transcendental, so this fact should probably be given without a proof. However, it is interesting why every line segment built from a segment of length 1 using only compass and straightedge has an algebraic length. I don't think I have ever studied a rigorous proof. However, this may be more of a geometrical than number-theoretical theorem.

Finally, I would look at public-key cryptography, but I don't know if it is possible to omit enough details to make it into both an accessible and a rigorous course.

- #4

Evgeny.Makarov

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From the title page:Fernando Revilla said:

An Introduction to Number Theory

"An Introduction to

Number Theory

With 16 Figures"

I wanted to exclaim, "Yeah, this is not geometry, baby! You should be happy with just a handful of pictures". (Smile) Also, the beginning of "Alice in Wonderland" comes to mind.

Alice was beginning to get very tired of sitting by her sister on the bank, and of having nothing to do: once or twice she had peeped into the book her sister was reading, but it had no pictures or conversations in it, `and what is the use of a book,' thought Alice `without pictures or conversation?'

- #5

Nono713

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Evgeny.Makarov said:Finally, I would look at public-key cryptography, but I don't know if it is possible to omit enough details to make it into both an accessible and a rigorous course.

I don't think so. It's fun and all as an example of applications of number theory but without computational complexity theory and an actual course in cryptography the students are going to be lost. Cryptography needs its own separate course IMHO

Also, on topic: modular arithmetic. The single most important tool. No elementary number theory course is complete without it.

Elementary number theory is a branch of mathematics that deals with the study of positive integers and their properties. It focuses on the fundamental concepts and principles of numbers, such as divisibility, prime numbers, and factorization.

The topics typically covered in a first elementary number theory course include basic number theory concepts such as divisibility, prime numbers, greatest common divisor, least common multiple, and modular arithmetic. Other topics may include Euclidean algorithm, prime factorization, and the fundamental theorem of arithmetic.

Studying elementary number theory helps develop critical thinking skills and problem-solving abilities. It also provides a foundation for understanding more advanced mathematical concepts and is applicable in fields such as cryptography, computer science, and data encryption.

Elementary number theory has many practical applications in our daily lives. It is used in fields such as banking and finance for secure data encryption, in computer science for coding and data compression, and in cryptography for ensuring secure communication and transactions.

There are many excellent resources available for learning elementary number theory, including textbooks, online courses, and interactive learning tools. Some recommended resources include "Elementary Number Theory" by Kenneth Rosen, "A Friendly Introduction to Number Theory" by Joseph Silverman, and "Number Theory Through Inquiry" by Marshall and Odell. Online resources such as Khan Academy and Brilliant also offer free courses on elementary number theory.

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