{"product_id":"solved-and-unsolved-problems-in-number-theory-9781470476458","title":"Solved and Unsolved Problems in Number Theory","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eThe investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cul\u003e\n\u003cli\u003eChapter I: From Perfect Numbers to the Quadratic Reciprocity Law: 1 Perfect numbers\u003c\/li\u003e\n\u003cli\u003e2 Euclid\u003c\/li\u003e\n\u003cli\u003e3 Euler's converse proved\u003c\/li\u003e\n\u003cli\u003e4 Euclid's algorithm\u003c\/li\u003e\n\u003cli\u003e5 Cataldi and others\u003c\/li\u003e\n\u003cli\u003e6 The prime number theorem\u003c\/li\u003e\n\u003cli\u003e7 Two useful theorems\u003c\/li\u003e\n\u003cli\u003e8 Fermat and others\u003c\/li\u003e\n\u003cli\u003e9 Euler's generalization proved\u003c\/li\u003e\n\u003cli\u003e10 Perfect numbers, II\u003c\/li\u003e\n\u003cli\u003e11 Euler and $M_{31}$\u003c\/li\u003e\n\u003cli\u003e12 Many conjectures and their interrelations\u003c\/li\u003e\n\u003cli\u003e13 Splitting the primes into equinumerous classes\u003c\/li\u003e\n\u003cli\u003e14 Euler's criterion formulated\u003c\/li\u003e\n\u003cli\u003e15 Euler's criterion proved\u003c\/li\u003e\n\u003cli\u003e16 Wilson's theorem\u003c\/li\u003e\n\u003cli\u003e17 Gauss's criterion\u003c\/li\u003e\n\u003cli\u003e18 The original Legendre symbol\u003c\/li\u003e\n\u003cli\u003e19 The reciprocity law\u003c\/li\u003e\n\u003cli\u003e20 The prime divisors of $n^2 +a$\u003c\/li\u003e\n\u003cli\u003eChapter II: The Underlying Structure: 21 The residue classes as an invention\u003c\/li\u003e\n\u003cli\u003e22 The residue classes as a tool\u003c\/li\u003e\n\u003cli\u003e23 The residue classes as a group\u003c\/li\u003e\n\u003cli\u003e24 Quadratic residues\u003c\/li\u003e\n\u003cli\u003e25 Is the quadratic reciprocity law a deep theorem?\u003c\/li\u003e\n\u003cli\u003e26 Congruential equations with a prime modulus\u003c\/li\u003e\n\u003cli\u003e27 Euler's $\\phi$ function\u003c\/li\u003e\n\u003cli\u003e28 Primitive roots with a prime modulus\u003c\/li\u003e\n\u003cli\u003e29 $\\mathfrak{M}_{p}$ as a cyclic group\u003c\/li\u003e\n\u003cli\u003e30 The circular parity switch\u003c\/li\u003e\n\u003cli\u003e31 Primitive roots and Fermat numbers\u003c\/li\u003e\n\u003cli\u003e32 Artin's conjectures\u003c\/li\u003e\n\u003cli\u003e33 Questions concerning cycle graphs\u003c\/li\u003e\n\u003cli\u003e34 Answers concerning cycle graphs\u003c\/li\u003e\n\u003cli\u003e35 Factor generators of $\\mathfrak{M}_{m}$\u003c\/li\u003e\n\u003cli\u003e36 Primes in some arithmetic progressions and a general divisibility theorem\u003c\/li\u003e\n\u003cli\u003e37 Scalar and vector indices\u003c\/li\u003e\n\u003cli\u003e38 The other residue classes\u003c\/li\u003e\n\u003cli\u003e39 The converse of Fermat's theorem\u003c\/li\u003e\n\u003cli\u003e40 Sufficient conditions for primality\u003c\/li\u003e\n\u003cli\u003eChapter III: Pythagoreanism and Its Many Consequences: 41 The Pythagoreans\u003c\/li\u003e\n\u003cli\u003e42 The Pythagorean theorem\u003c\/li\u003e\n\u003cli\u003e43 The $\\sqrt 2$ and the crisis\u003c\/li\u003e\n\u003cli\u003e44 The effect upon geometry\u003c\/li\u003e\n\u003cli\u003e45 The case for Pythagoreanism\u003c\/li\u003e\n\u003cli\u003e46 Three Greek problems\u003c\/li\u003e\n\u003cli\u003e47 Three theorems of Fermat\u003c\/li\u003e\n\u003cli\u003e48 Fermat's last ``Theorem''\u003c\/li\u003e\n\u003cli\u003e49 The easy case and infinite descent\u003c\/li\u003e\n\u003cli\u003e50 Gaussian integers and two applications\u003c\/li\u003e\n\u003cli\u003e51 Algebraic integers and Kummer's theorem\u003c\/li\u003e\n\u003cli\u003e52 The restricted case, Sophie Germain, and Wieferich\u003c\/li\u003e\n\u003cli\u003e53 Euler's ``Conjecture''\u003c\/li\u003e\n\u003cli\u003e54 Sum of two squares\u003c\/li\u003e\n\u003cli\u003e55 A generalization and geometric number theory\u003c\/li\u003e\n\u003cli\u003e56 A generalization and binary quadratic forms\u003c\/li\u003e\n\u003cli\u003e57 Some applications\u003c\/li\u003e\n\u003cli\u003e58 The significance of Fermat's equation\u003c\/li\u003e\n\u003cli\u003e59 The main theorem\u003c\/li\u003e\n\u003cli\u003e60 An algorithm\u003c\/li\u003e\n\u003cli\u003e61 Continued fractions for $\\sqrt N$\u003c\/li\u003e\n\u003cli\u003e62 From Archimedes to Lucas\u003c\/li\u003e\n\u003cli\u003e63 The Lucas criterion\u003c\/li\u003e\n\u003cli\u003e64 A probability argument\u003c\/li\u003e\n\u003cli\u003e65 Fibonacci numbers and the original Lucas test\u003c\/li\u003e\n\u003cli\u003eAppendix to Chapters I-III: Supplementary comments, theorems, and exercises\u003c\/li\u003e\n\u003cli\u003eChapter IV: Progress: 66 Chapter I fifteen years later\u003c\/li\u003e\n\u003cli\u003e67 Artin's conjectures, II\u003c\/li\u003e\n\u003cli\u003e68 Cycle graphs and related topics\u003c\/li\u003e\n\u003cli\u003e69 Pseudoprimes and primality\u003c\/li\u003e\n\u003cli\u003e70 Fermat's last ``Theorem,'' II\u003c\/li\u003e\n\u003cli\u003e71 Binary quadratic forms with negative discriminants\u003c\/li\u003e\n\u003cli\u003e72 Binary quadratic forms with positive discriminants\u003c\/li\u003e\n\u003cli\u003e73 Lucas and Pythagoras\u003c\/li\u003e\n\u003cli\u003e74 The progress report concluded\u003c\/li\u003e\n\u003cli\u003e75 The second progress report begins\u003c\/li\u003e\n\u003cli\u003e76 On judging conjectures\u003c\/li\u003e\n\u003cli\u003e77 On judging conjectures, II\u003c\/li\u003e\n\u003cli\u003e78 Subjective judgement, the creation of conjectures and inventions\u003c\/li\u003e\n\u003cli\u003e79 Fermat's last ``Theorem,'' III\u003c\/li\u003e\n\u003cli\u003e80 Computing and algorithms\u003c\/li\u003e\n\u003cli\u003e81 $\\scr{C}(3)\\times\\scr{C}(3)\\times\\scr{C}(3)\\times\\scr{C}(3)$ and all that\u003c\/li\u003e\n\u003cli\u003e82 1993\u003c\/li\u003e\n\u003cli\u003eAppendix: Statement on fundamentals\u003c\/li\u003e\n\u003cli\u003eTable of definitions\u003c\/li\u003e\n\u003cli\u003eReferences\u003c\/li\u003e\n\u003cli\u003eIndex.\u003c\/li\u003e\n\u003c\/ul\u003e","brand":"American Mathematical Society","offers":[{"title":"Default Title","offer_id":51138281505111,"sku":"9781470476458","price":52.2,"currency_code":"GBP","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781470476458.jpg?v=1751918724","url":"https:\/\/bookcurl.com\/products\/solved-and-unsolved-problems-in-number-theory-9781470476458","provider":"Book Curl","version":"1.0","type":"link"}