國立虎尾科技大學 |

A Pythagorean Introduction to Number Theory = Right Triangles, Sums of Squares, and Arithmetic /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	A Pythagorean Introduction to Number Theory/ by Ramin Takloo-Bighash.
Reminder of title:	Right Triangles, Sums of Squares, and Arithmetic /
Author:	Takloo-Bighash, Ramin.
Description:	XVIII, 279 p. 24 illus., 9 illus. in color.online resource. :
Contained By:	Springer Nature eBook
Subject:	Number theory. -
Online resource:	https://doi.org/10.1007/978-3-030-02604-2
ISBN:	9783030026042

A Pythagorean Introduction to Number Theory = Right Triangles, Sums of Squares, and Arithmetic /
Takloo-Bighash, Ramin.

A Pythagorean Introduction to Number TheoryRight Triangles, Sums of Squares, and Arithmetic /[electronic resource] :by Ramin Takloo-Bighash. - 1st ed. 2018. - XVIII, 279 p. 24 illus., 9 illus. in color.online resource. - Undergraduate Texts in Mathematics,0172-6056. - Undergraduate Texts in Mathematics,.

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.

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.

ISBN: 9783030026042

Standard No.: 10.1007/978-3-030-02604-2doiSubjects--Topical Terms:

527883
Number theory.

LC Class. No.: QA241-247.5

Dewey Class. No.: 512.7

A Pythagorean Introduction to Number Theory = Right Triangles, Sums of Squares, and Arithmetic /
LDR:03643nam a22003975i 4500 001 987705
003 DE-He213
005 20200630133203.0
007 cr nn 008mamaa
008 201225s2018 gw | s |||| 0|eng d
020 $a 9783030026042 $9 978-3-030-02604-2
024 7 $a 10.1007/978-3-030-02604-2 $2 doi
035 $a 978-3-030-02604-2
050 4 $a QA241-247.5
072 7 $a PBH $2 bicssc
072 7 $a MAT022000 $2 bisacsh
072 7 $a PBH $2 thema
082 0 4 $a 512.7 $2 23
100 1 $a Takloo-Bighash, Ramin. $4 aut $4 http://id.loc.gov/vocabulary/relators/aut $3 1211334
245 1 2 $a A Pythagorean Introduction to Number Theory $h [electronic resource] : $b Right Triangles, Sums of Squares, and Arithmetic / $c by Ramin Takloo-Bighash.
250 $a 1st ed. 2018.
264 1 $a Cham : $b Springer International Publishing : $b Imprint: Springer, $c 2018.
300 $a XVIII, 279 p. 24 illus., 9 illus. in color. $b online resource.
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
347 $a text file $b PDF $2 rda
490 1 $a Undergraduate Texts in Mathematics, $x 0172-6056
505 0 $a 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.
520 $a 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.
650 0 $a Number theory. $3 527883
650 1 4 $a Number Theory. $3 672023
710 2 $a SpringerLink (Online service) $3 593884
773 0 $t Springer Nature eBook
776 0 8 $i Printed edition: $z 9783030026035
776 0 8 $i Printed edition: $z 9783030026059
830 0 $a Undergraduate Texts in Mathematics, $x 0172-6056 $3 1253653
856 4 0 $u https://doi.org/10.1007/978-3-030-02604-2
912 $a ZDB-2-SMA
912 $a ZDB-2-SXMS
950 $a Mathematics and Statistics (SpringerNature-11649)
950 $a Mathematics and Statistics (R0) (SpringerNature-43713)