LEADER 05168nam 22008175 450 001 9910254088703321 005 20241023170746.0 010 $a3-319-43875-1 024 7 $a10.1007/978-3-319-43875-7 035 $a(CKB)3710000000872820 035 $a(DE-He213)978-3-319-43875-7 035 $a(MiAaPQ)EBC6310586 035 $a(MiAaPQ)EBC5588862 035 $a(Au-PeEL)EBL5588862 035 $a(OCoLC)1066193579 035 $a(PPN)195513886 035 $a(EXLCZ)993710000000872820 100 $a20160919d2016 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aNumber Theory $eAn Introduction via the Density of Primes /$fby Benjamin Fine, Gerhard Rosenberger 205 $a2nd ed. 2016. 210 1$aCham :$cSpringer International Publishing :$cImprint: Birkhäuser,$d2016. 215 $a1 online resource (XIII, 413 p. 12 illus., 1 illus. in color.) 311 $a3-319-43873-5 320 $aIncludes bibliographical references and index. 327 $aIntroduction and Historical Remarks -- Basic Number Theory -- The Infinitude of Primes -- The Density of Primes -- Primality Testing: An Overview -- Primes and Algebraic Number Theory -- The Fields Q_p of p-adic Numbers: Hensel's Lemma -- References -- Index. 330 $aNow in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is one of polynomial time, a topic not usually included in such texts Many interesting ancillary topics, such as primality testing and cryptography, Fermat and Mersenne numbers, and Carmichael numbers The user-friendly style, historical context, and wide range of exercises that range from simple to quite difficult (with solutions and hints provided for select exercises) make Number Theory: An Introduction via the Density of Primes ideal for both self-study and classroom use. Intended for upper level undergraduates and beginning graduates, the only prerequisites are a basic knowledge of calculus, multivariable calculus, and some linear algebra. All necessary concepts from abstract algebra and complex analysis are introduced where needed. 606 $aNumber theory 606 $aLogic, Symbolic and mathematical 606 $aMatrix theory 606 $aAlgebra 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aApplied mathematics 606 $aEngineering mathematics 606 $aData structures (Computer science) 606 $aNumber Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M25001 606 $aMathematical Logic and Foundations$3https://scigraph.springernature.com/ontologies/product-market-codes/M24005 606 $aLinear and Multilinear Algebras, Matrix Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/M11094 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 606 $aApplications of Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M13003 606 $aData Structures and Information Theory$3https://scigraph.springernature.com/ontologies/product-market-codes/I15009 615 0$aNumber theory. 615 0$aLogic, Symbolic and mathematical. 615 0$aMatrix theory. 615 0$aAlgebra. 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 0$aApplied mathematics. 615 0$aEngineering mathematics. 615 0$aData structures (Computer science) 615 14$aNumber Theory. 615 24$aMathematical Logic and Foundations. 615 24$aLinear and Multilinear Algebras, Matrix Theory. 615 24$aAnalysis. 615 24$aApplications of Mathematics. 615 24$aData Structures and Information Theory. 676 $a512.7 700 $aFine$b Benjamin$4aut$4http://id.loc.gov/vocabulary/relators/aut$056763 702 $aRosenberger$b Gerhard$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254088703321 996 $aNumber Theory$92039114 997 $aUNINA