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A Pythagorean Introduction to Number Theory

Right Triangles, Sums of Squares, and Arithmetic

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Publisher:

Springer

Pages: 279
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Recommend to library


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Hardcover - 9783030026035

10 December 2018

$54.99

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Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about...

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Right triangles are at the heart of this textbook’s vibrant new approach to elementary number theory. Inspired by the familiar Pythagorean theorem, the author invites the reader to ask natural arithmetic questions about right triangles, then proceeds to develop the theory needed to respond. Throughout, students are encouraged to engage with the material by posing questions, working through exercises, using technology, and learning about the broader context in which ideas developed.

Progressing from the fundamentals of number theory through to Gauss sums and quadratic reciprocity, the first part of this text presents an innovative first course in elementary number theory. The advanced topics that follow, such as counting lattice points and the four squares theorem, offer a variety of options for extension, or a higher-level course; the breadth and modularity of the later material is ideal for creating a senior capstone course. Numerous exercises are included throughout, many of which are designed for SageMath.

By involving students in the active process of inquiry and investigation, this textbook imbues the foundations of number theory with insights into the lively mathematical process that continues to advance the field today. Experience writing proofs is the only formal prerequisite for the book, while a background in basic real analysis will enrich the reader’s appreciation of the final chapters.

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Offers an innovative approach to elementary number theory motivated by right triangles

Inspires students to explore number theory through investigation of concrete examples

Provides historical context throughout, showing how ideas developed in the field

Features numerous engaging exercises, including many designed for SageMath

Part I Foundational Material
1. Introduction
2. Basic number theory
3. Integral solutions to the Pythagorean Equation
4. What integers are areas of right triangles?
5. What numbers are the edges of a right triangle?
6. Primes of the form 4k+1
7. Gauss sums, Quadratic Reciprocity, and the Jacobi symbol
Part II Advanced Topics
8. Counting Pythagorean triples modulo an integer
9. How many lattice points are there on a circle or a sphere?
10. What about geometry?
11. Another proof of the four squares theorem
12. Quadratic forms and sums of squares
13. How many Pythagorean triples are there?
14. How are rational points distributed, really?
Part III Appendices
A. Background
B. Algebraic integers
C. SageMath
References
Index.
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Ramin Takloo-Bighash is a Professor of Mathematics at the University of Illinois at Chicago, where his research is centered on Diophantine geometry and automorphic forms. His enthusiasm for research inspires his teaching at all levels.

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Ramin Takloo-Bighash is a Professor of Mathematics at the University of Illinois at Chicago, where his research is centered on Diophantine geometry and automorphic forms. His enthusiasm for research inspires his teaching at all levels.

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