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001 978-3-030-02604-2
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020 _a9783030026042
_9978-3-030-02604-2
024 7 _a10.1007/978-3-030-02604-2
_2doi
050 4 _aQA241-247.5
072 7 _aPBH
_2bicssc
072 7 _aMAT022000
_2bisacsh
072 7 _aPBH
_2thema
082 0 4 _a512.7
_223
100 1 _aTakloo-Bighash, Ramin.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
245 1 2 _aA Pythagorean Introduction to Number Theory
_h[electronic resource] :
_bRight Triangles, Sums of Squares, and Arithmetic /
_cby Ramin Takloo-Bighash.
250 _a1st ed. 2018.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2018.
300 _aXVIII, 279 p. 24 illus., 9 illus. in color.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aUndergraduate Texts in Mathematics,
_x0172-6056
505 0 _aPart 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 _aRight 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 _aNumber theory.
650 1 4 _aNumber Theory.
_0http://scigraph.springernature.com/things/product-market-codes/M25001
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783030026035
776 0 8 _iPrinted edition:
_z9783030026059
830 0 _aUndergraduate Texts in Mathematics,
_x0172-6056
856 4 0 _uhttps://doi.org/10.1007/978-3-030-02604-2
912 _aZDB-2-SMA
999 _c17692
_d17692
942 _cebook