{"product_id":"elementary-number-theory-with-programming-isbn-9781119062769","title":"Elementary Number Theory with Programming","description":"\u003cp\u003e\u003cb\u003eA highly successful presentation of the fundamental concepts of number theory and computer programming\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eBridging an existing gap between mathematics and programming, \u003ci\u003eElementary Number Theory with Programming \u003c\/i\u003eprovides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and concepts in either area.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eElementary Number Theory with Programming \u003c\/i\u003efeatures comprehensive coverage of the methodology and applications of the most well-known theorems, problems, and concepts in number theory. Using standard mathematical applications within the programming field, the book presents modular arithmetic and prime decomposition, which are the basis of the public-private key system of cryptography. In addition, the book includes:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eNumerous examples, exercises, and research challenges in each chapter to encourage readers to work through the discussed concepts and ideas\u003c\/li\u003e \u003cli\u003eSelect solutions to the chapter exercises in an appendix\u003c\/li\u003e \u003cli\u003ePlentiful sample computer programs to aid comprehension of the presented material for readers who have either never done any programming or need to improve their existing skill set\u003c\/li\u003e \u003cli\u003eA related website with links to select exercises\u003c\/li\u003e \u003cli\u003eAn Instructor’s Solutions Manual available on a companion website\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eElementary Number Theory with Programming \u003c\/i\u003eis a useful textbook for undergraduate and graduate-level students majoring in mathematics or computer science, as well as an excellent supplement for teachers and students who would like to better understand and appreciate number theory and computer programming. The book is also an ideal reference for computer scientists, programmers, and researchers interested in the mathematical applications of programming.\u003c\/p\u003e \u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003eWords xiii\u003c\/p\u003e \u003cp\u003eNotation in Mathematical Writing and in Programming xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Special Numbers: Triangular, Oblong, Perfect, Deficient, and Abundant 1\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include one for factoring numbers and one to test a conjecture up to a fixed limit.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eTriangular Numbers 1\u003c\/p\u003e \u003cp\u003eOblong Numbers and Squares 3\u003c\/p\u003e \u003cp\u003eDeficient, Abundant, and Perfect Numbers 4\u003c\/p\u003e \u003cp\u003eExercises 7\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Fibonacci Sequence, Primes, and the Pell Equation 13\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include examples that count steps to compare two different approaches.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003ePrime Numbers and Proof by Contradiction 13\u003c\/p\u003e \u003cp\u003eProof by Construction 17\u003c\/p\u003e \u003cp\u003eSums of Two Squares 18\u003c\/p\u003e \u003cp\u003eBuilding a Proof on Prior Assertions 18\u003c\/p\u003e \u003cp\u003eSigma Notation 19\u003c\/p\u003e \u003cp\u003eSome Sums 19\u003c\/p\u003e \u003cp\u003eFinding Arithmetic Functions 20\u003c\/p\u003e \u003cp\u003eFibonacci Numbers 22\u003c\/p\u003e \u003cp\u003eAn Infinite Product 26\u003c\/p\u003e \u003cp\u003eThe Pell Equation 26\u003c\/p\u003e \u003cp\u003eGoldbach’s Conjecture 30\u003c\/p\u003e \u003cp\u003eExercises 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Pascal’s Triangle 44\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include examples that generate factorial using iteration and using recursion and thus demonstrate and compare important techniques in programming.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eFactorials 44\u003c\/p\u003e \u003cp\u003eThe Combinatorial Numbers \u003ci\u003en \u003c\/i\u003eChoose \u003ci\u003ek \u003c\/i\u003e46\u003c\/p\u003e \u003cp\u003ePascal’s Triangle 48\u003c\/p\u003e \u003cp\u003eBinomial Coefficients 50\u003c\/p\u003e \u003cp\u003eExercises 50\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Divisors and Prime Decomposition 56\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include one that uses the algorithm to produce the GCD of a pair of numbers and a program to produce the prime decomposition of a number.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eDivisors 56\u003c\/p\u003e \u003cp\u003eGreatest Common Divisor 58\u003c\/p\u003e \u003cp\u003eDiophantine Equations 65\u003c\/p\u003e \u003cp\u003eLeast Common Multiple 67\u003c\/p\u003e \u003cp\u003ePrime Decomposition 68\u003c\/p\u003e \u003cp\u003eSemiprime Numbers 70\u003c\/p\u003e \u003cp\u003eWhen is a Number an \u003ci\u003em\u003c\/i\u003eth Power? 71\u003c\/p\u003e \u003cp\u003eTwin Primes 73\u003c\/p\u003e \u003cp\u003eFermat Primes 73\u003c\/p\u003e \u003cp\u003eOdd Primes Are Differences of Squares 74\u003c\/p\u003e \u003cp\u003eWhen is \u003ci\u003en \u003c\/i\u003ea Linear Combination of \u003ci\u003ea \u003c\/i\u003eand \u003ci\u003eb\u003c\/i\u003e? 75\u003c\/p\u003e \u003cp\u003ePrime Decomposition of \u003ci\u003en\u003c\/i\u003e! 76\u003c\/p\u003e \u003cp\u003eNo Nonconstant Polynomial with Integer Coefficients Assumes Only Prime Values 77\u003c\/p\u003e \u003cp\u003eExercises 78\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Modular Arithmetic 85\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eOne program checks if a mod equation is true, and another determines the solvability of a mod equation and then solves an equation that is solvable by a brute-force approach.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eCongruence Classes Mod \u003ci\u003ek \u003c\/i\u003e85\u003c\/p\u003e \u003cp\u003eLaws of Modular Arithmetic 87\u003c\/p\u003e \u003cp\u003eModular Equations 90\u003c\/p\u003e \u003cp\u003eFermat’s Little Theorem 91\u003c\/p\u003e \u003cp\u003eFermat’s Little Theorem 92\u003c\/p\u003e \u003cp\u003eMultiplicative Inverses 92\u003c\/p\u003e \u003cp\u003eWilson’s Theorem 93\u003c\/p\u003e \u003cp\u003eWilson’s Theorem 95\u003c\/p\u003e \u003cp\u003eWilson’s Theorem (2nd Version) 95\u003c\/p\u003e \u003cp\u003eSquares and Quadratic Residues 96\u003c\/p\u003e \u003cp\u003eLagrange’s Theorem 98\u003c\/p\u003e \u003cp\u003eLagrange’s Theorem 99\u003c\/p\u003e \u003cp\u003eReduced Pythagorean Triples 100\u003c\/p\u003e \u003cp\u003eChinese Remainder Theorem 102\u003c\/p\u003e \u003cp\u003eChinese Remainder Theorem 103\u003c\/p\u003e \u003cp\u003eExercises 104\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Number Theoretic Functions 111\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include two distinct approaches to calculating the tau function.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eThe \u003ci\u003eTau \u003c\/i\u003eFunction 111\u003c\/p\u003e \u003cp\u003eThe \u003ci\u003eSigma \u003c\/i\u003eFunction 114\u003c\/p\u003e \u003cp\u003eMultiplicative Functions 115\u003c\/p\u003e \u003cp\u003ePerfect Numbers Revisited 115\u003c\/p\u003e \u003cp\u003eMersenne Primes 116\u003c\/p\u003e \u003cp\u003e\u003ci\u003eF\u003c\/i\u003e(\u003ci\u003en\u003c\/i\u003e) = Σ\u003ci\u003ef\u003c\/i\u003e(\u003ci\u003ed\u003c\/i\u003e) Where \u003ci\u003ed \u003c\/i\u003eis a Divisor of \u003ci\u003en \u003c\/i\u003e117\u003c\/p\u003e \u003cp\u003eThe Möbius Function 119\u003c\/p\u003e \u003cp\u003eThe Riemann Zeta Function 121\u003c\/p\u003e \u003cp\u003eExercises 124\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 The Euler Phi Function 134\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs demonstrate two approaches to calculating the phi function.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eThe \u003ci\u003ePhi \u003c\/i\u003eFunction 134\u003c\/p\u003e \u003cp\u003eEuler’s Generalization of Fermat’s Little Theorem 138\u003c\/p\u003e \u003cp\u003ePhi of a Product of \u003ci\u003em \u003c\/i\u003eand \u003ci\u003en \u003c\/i\u003eWhen \u003ci\u003egcd\u003c\/i\u003e(\u003ci\u003em\u003c\/i\u003e,\u003ci\u003en\u003c\/i\u003e) \u0026gt; 1 139\u003c\/p\u003e \u003cp\u003eThe Order of \u003ci\u003ea \u003c\/i\u003e(\u003ci\u003emod n\u003c\/i\u003e) 139\u003c\/p\u003e \u003cp\u003ePrimitive Roots 140\u003c\/p\u003e \u003cp\u003eThe Index of \u003ci\u003em \u003c\/i\u003e(\u003ci\u003emod p\u003c\/i\u003e) Relative to \u003ci\u003ea \u003c\/i\u003e141\u003c\/p\u003e \u003cp\u003eTo Be or Not to Be a Quadratic Residue 145\u003c\/p\u003e \u003cp\u003eThe Legendre Symbol 146\u003c\/p\u003e \u003cp\u003eQuadratic Reciprocity 147\u003c\/p\u003e \u003cp\u003eLaw of Quadratic Reciprocity 148\u003c\/p\u003e \u003cp\u003eWhen Does \u003ci\u003ex\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e = \u003ci\u003ea \u003c\/i\u003e(\u003ci\u003emod n\u003c\/i\u003e) Have a Solution? 148\u003c\/p\u003e \u003cp\u003eExercises 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Sums and Partitions 158\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe exposition explains the central role of binary representation in computing and the programs produce the binary partition using a built-in function.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eAn \u003ci\u003en\u003c\/i\u003eth Power is the Sum of Two Squares 158\u003c\/p\u003e \u003cp\u003eSolutions to the Diophantine Equation \u003ci\u003ea\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e + \u003ci\u003eb\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e + \u003ci\u003ec\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e = \u003ci\u003ed\u003c\/i\u003e\u003csup\u003e2\u003c\/sup\u003e 159\u003c\/p\u003e \u003cp\u003eRow Sums of a Triangular Array of Consecutive Odd Numbers 160\u003c\/p\u003e \u003cp\u003ePartitions 160\u003c\/p\u003e \u003cp\u003eWhen is a Number the Sum of Two Squares? 167\u003c\/p\u003e \u003cp\u003eSums of Four or Fewer Squares 170\u003c\/p\u003e \u003cp\u003eExercises 175\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Cryptography 182\u003cbr\u003e\u003c\/b\u003e\u003ci\u003eThe programs include different ways to generate counts of letters and also Fermat factoring.\u003c\/i\u003e\u003c\/p\u003e \u003cp\u003eIntroduction and History 182\u003c\/p\u003e \u003cp\u003ePublic-Key Cryptography 187\u003c\/p\u003e \u003cp\u003eFactoring Large Numbers 188\u003c\/p\u003e \u003cp\u003eThe Knapsack Problem 191\u003c\/p\u003e \u003cp\u003eSuperincreasing Sequences 192\u003c\/p\u003e \u003cp\u003eExercises 194\u003c\/p\u003e \u003cp\u003eAnswers or Hints to Selected Exercises 203\u003c\/p\u003e \u003cp\u003eIndex 207\u003c\/p\u003e \"It consists of nine chapters, all including the corresponding programs along with their mathematical content. The mathematical structure is also interesting and well-formed starting from special numbers, primes and Pell equation, to Pascal's triangle, prime decomposition and modular arithmetic and finishing with number-theoretic functions, the Euler Phi-function, sums and partitions and the classical application to cryptography. It is also remarkable that the main scope of the programs is defined before their use from the reader, providing him the best orientation for his study.\" (Zentralblatt MATH 2016) \u003cp\u003e\u003cb\u003eMarty Lewinter\u003c\/b\u003e, PhD, is Professor Emeritus of Mathematics at the State University of New York, Purchase College. The author of three books and more than 80 articles, he is Executive Director of Mathematics at American Digital University Services.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eJeanine Meyer\u003c\/b\u003e, PhD, is Professor of Mathematics\/Computer Science at the State University of New York, Purchase College. She is the author of six books as well as numerous journal articles.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA successful presentation of the fundamental concepts of number theory and computer programming\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eBridging an existing gap between mathematics and programming, \u003ci\u003eElementary Number Theory with Programming\u003c\/i\u003e provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and concepts in either area.\u003c\/p\u003e \u003ci\u003eElementary Number Theory with Programming\u003c\/i\u003e features comprehensive coverage of the methodology and applications of the most well-known theorems, problems, and concepts in number theory. Using standard mathematical applications within the programming field, the book presents triangle numbers and prime decomposition, which are the basis of the public-private key system of cryptography. In addition, the book includes:\u003cbr\u003e \u003cul\u003e \u003cli\u003eNumerous examples,  exercises, and research challenges in each chapter to encourage readers to work through the discussed concepts and ideas \u003c\/li\u003e \u003cli\u003eSelect solutions to the chapter exercises in an appendix\u003c\/li\u003e \u003cli\u003ePlentiful sample computer programs to aid comprehension of the presented material for readers who have either never done any programming or need to improve their existing skill set\u003c\/li\u003e \u003cli\u003e A related website with links to select exercises\u003c\/li\u003e \u003cli\u003eAn Instructor’s Solutions Manual available on a companion website\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eElementary Number Theory with Programming\u003c\/i\u003e is a useful textbook for undergraduate and graduate-level students majoring in mathematics or computer science, as well as an excellent supplement for teachers and students who would like to better understand and appreciate number theory and computer programming. The book is also an ideal reference for computer scientists, programmers, and researchers interested in the mathematical applications of programming.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eMarty Lewinter, PhD,\u003c\/b\u003e is Professor Emeritus of Mathematics at the State University of New York, Purchase College. The author of three books and more than 80 articles, he is Executive Director of Mathematics at American Digital University Services.\u003c\/p\u003e \u003cp\u003e\u003cb\u003eJeanine Meyer, PhD, \u003c\/b\u003eis Professor of Computer Science at the State University of New York, Purchase College. She is the author of six books as well as numerous journal articles. \u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989118828773,"sku":"NP9781119062769","price":76.5,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9781119062769.jpg?v=1761782872","url":"https:\/\/k12savings.com\/products\/elementary-number-theory-with-programming-isbn-9781119062769","provider":"K12savings","version":"1.0","type":"link"}