##
**Advanced number theory with applications.**
*(English)*
Zbl 1200.11002

Discrete Mathematics and Its Applications. Boca Raton, FL: CRC Press (ISBN 978-1-4200-8328-6/hbk; 978-1-138-11325-1/pbk; 978-1-4200-8329-3/ebook). xiii, 466 p. (2010).

Richard A. Mollin, a renowned and popular author of numerous textbooks in number theory and its allied areas of mathematics, presents here his most recent treatise on the subject. As he points out in the preface, the current book is designed as a second course in number theory at the senior undergraduate or junior graduate level to follow a foregoing course in elementary methods, such as that given in his previous textbook “Fundamental Number Theory with Applications” [CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press (1998; Zbl 0943.11001)]. Actually, the reader of the present text is assumed to have a profound knowledge of the contents of this foregoing book, and the author abundantly refers to those throughout the entire exposition. As for the contents of the book under review, the material is organized in ten chapters, each of which is subdivided into several sections.

Chapter 1 is entitled “Algebraic Number Theory and Quadratic Fields” and provides the necessary basics on algebraic number fields,with a special view to quadratic fields and their applications to solutions of the Ramanujan–Nagell Diophantine equations, factorization of Gaussian integers, Euclidean quadratic fields, and Fermat’s Last Theorem for the exponent \(p= 3\). Further applications of the unique factorization property concern classical solutions of Bachet’s Diophantine equation as well as a characterization of norm-Euclidean quadratic fields.

Chapter 2 discusses the ideal theory of rings of integers in quadratic number fields, together with the relevant algebraic background material on Noetherian domains, Dedekind domains, and factorial rings. As for concrete applications, factoring methods for certain cubic integers are illustrated by means of Pollard’s method and higher Fermat numbers.

Chapter 3 studies binary quadratic forms from various points of view. Starting with the basic material on equivalence, discriminants, reduction, and class numbers, the author proceeds by comparing form class numbers and ideal class numbers, giving an alternative proof of the finiteness of the ideal class number, investigating the concept of ambiguity for forms and ideals, analyzing the genus of forms, and describing the assigned values of generic characters via Jacobi symbols. At the end of this chapter, the foregoing methods and results are applied to give numerous examples of representations of primes in the form \(p= a^2+ Db^2\), on the one hand, and to illustrate the equivalence of forms modulo a prime number \(p\) on the other.

Chapter 4 is devoted to the topic of Diophantine approximation, thereby assuming the elementary background on continued fractions, rational approximations, quadratic rationals, and related topics. Algebraic and transcendental numbers, the Thue-Siegel-Roth Theorem, Schanuel’s Conjecture, and a brief discussion of the famous number-theoretic constants of Gel’fond, Gel’fond-Schneider, Prouhet-Thue-Morse, Euler, Apéry, and Catalan are the main themes taken up here, along with a section on the relevant material from Minkowski’s geometry of numbers.

Chapter 5 provides some extended, more advanced knowledge of arithmetic functions, including Bernoulli numbers, Bernoulli polynomials, Fourier series, the Euler-Maclaurin summation formula, Wallis’s and Stirling’s formula, and the accurate approximation of the Euler-Mascheroni constant. Then, after a subsequent section on average orders of arithmetic functions and some concrete applications, the Riemann zeta function is briefly explained, in particular with a view toward its relations to the celebrated Prime Number Theorem and its various arithmetic function equivalences, on the one hand, and Riemann’s hypothesis on the other.

Chapter 6 turns to the methods of \(p\)-adic analysis in number theory, with the focus on Hensel’s Lemma, valuation theory for fields, Ostrowski’s Theorem, and the representation of \(p\)-adic numbers as power series.

The principal goal of Chapter 7 is to establish Dirichlet’s Theorem on prime numbers in arithmetic progressions. To this end, the author first discusses Dirichlet characters, Dirichlet \(L\)-functions, and the related conjectural “Generalized Riemann Hypothesis”. After a proof of Dirichlet’s Theorem along these lines, Dirichlet densities and some of their applications are exhibited, culminating in the theorem on Dirichlet density of primes in arithmetic progressions.

Chapter 8 gives several applications of the previous chapters to Diophantine equations. In the first part, Lucas-Lehmer functions and their various identities are used to analyze the solutions of the classical “generalized Ramanujan-Nagell equations” \(x^2- D= p^n\) and those of the Bachet equations \(y^2= x^3+ k\) thereby completing the discussion begun in Chapter 1. This is followed by a thorough treatment of Fermat’s Last Theorem for regular prime exponents, including Kummer’s pioneering proof of it, whereas the last section of this chapter is devoted to both the spectacular “ABC Conjecture” and the former “Catalan Conjecture”. The author depicts the present state of art regarding these milestones in number theory, explains the still unresolved “Fermat–Catalan Conjecture” in this context, and turns then to the crucial significance of the ABC Conjecture. Several consequences of the latter are derived, among them being the Thue-Siegel-Roth Theorem, Hall’s Conjecture, the Erdős-Mollin-Walsh Conjecture, the Granville-Langevin Conjecture, and other related (conjectural) statements.

Chapter 9 introduces elliptic curves over an arbitrary field, mainly with a view toward their arithmetic properties and applications. After displaying the basics, torsion points on elliptic curves, the Lutz-Nagell Theorem, Mazur’s Theorem, Siegel’s Theorem, and the reduction of rationals on elliptic curves in a survey-like manner, some recent applications are briefly described. These include H. W. Lenstra’s elliptic curve factoring method for odd composite natural numbers, his related elliptic curve primality test, and the effective primality proving algorithm by S. Goldwasser and J. Kilian [Proc. 18th STOC, 316–329 (1986)]. A brief section on elliptic curve cryptography, meant as a further application, concludes this chapter, with the focus on the Menezes-Okamoto-Vanstone elliptic curve cryptosystem (1991).

Chapter 10, the last chapter of the book, provides a concise introduction to modular forms, in general, and to elliptic modular functions in particular. This serves as the necessary background material for explaining the celebrated Shimura–Taniyama–Weil Conjecture both in terms of \(L\)-functions and modular representations. Using Ribet’s Theorem, it is demonstrated how this conjecture, which in 2001 has become what is now called the “Modularity Theorem”, implies the statement of Fermat’s Last Theorem.

Another central topic in number theory, namely sieve methods used to estimate the cardinalities of various sets of numbers defined by multiplicative properties, is treated in an appendix. Without proofs, the author provides here an overview of some classical open problems for which the use of sieve methods has led to remarkable recent advances. Along the way, the reader gets acquainted with Eratosthenes’s sieve in terms of the Möbius function, Brun’s constant, Selberg’s sieve, the Brun-Titchmarsh Theorem, Selberg’s sieve on twin primes, Selberg’s sieve on the Goldbach Conjecture, Artin’s Conjecture and the large sieve on it, the Bombieri-Vinogradov Theorem, the more recent Friedlander-Iwaniec Theorem, the Elliott-Halberstam Conjecture, and the nearly brand-new theorems by Goldston-Pintz-Yildirim on primes in tuples [cf. D. A. Goldston, J. Pintz and C. Y. Yildirim, The path to recent progress on small gaps between primes, Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany 2005. Providence, RI: American Mathematical Society (AMS). Clay Mathematics Proceedings 7, 129–139 (2007; Zbl 1213.11168)].

Finally, with these results as an illustration of the power of sieve methods, the use of the latter in tackling factorization problems is demonstrated by describing the powerful “Number Field Sieve Algorithm” due to Pollard, Buhler, H. W. Lenstra, Pomerance, and others.

In this context, the factoring of the ninth Fermat number is carried out as an instructive example.

As in the author’s foregoing, more elementary textbook “Fundamental Number Theory with Applications” [CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press (1998; Zbl 0943.11001)], each section comes with a large number of illustrating examples and accompanying exercises. Solutions of the odd-numbered exercises are included at the end of the text, and a solutions manual for the even-numbered exercises is available to instructors who adopt the text for a course. Also, there are about 50 biographical sketches of mathematicians who contributed to the development of the theories depicted in the course of the text, and each section features an introductory quotation of a historical celebrity as a cultural appetizer. The rich bibliography contains 106 references, where maximum information is imparted by explicit page reference for each citing of a given item within the text. The carefully compiled index has more than 1,500 entries presented for maximum cross-referencing. Overall, this excellent textbook bespeaks the author’s outstanding expository mastery just as much as his mathematical erudition and elevated taste. Presenting a wide panorama of topics in advanced classical and contemporary number theory, and that in an utmost lucid and comprehensible style of writing, the author takes the reader to the forefront of research in the field, and on a truly exciting journey over and above.

Chapter 1 is entitled “Algebraic Number Theory and Quadratic Fields” and provides the necessary basics on algebraic number fields,with a special view to quadratic fields and their applications to solutions of the Ramanujan–Nagell Diophantine equations, factorization of Gaussian integers, Euclidean quadratic fields, and Fermat’s Last Theorem for the exponent \(p= 3\). Further applications of the unique factorization property concern classical solutions of Bachet’s Diophantine equation as well as a characterization of norm-Euclidean quadratic fields.

Chapter 2 discusses the ideal theory of rings of integers in quadratic number fields, together with the relevant algebraic background material on Noetherian domains, Dedekind domains, and factorial rings. As for concrete applications, factoring methods for certain cubic integers are illustrated by means of Pollard’s method and higher Fermat numbers.

Chapter 3 studies binary quadratic forms from various points of view. Starting with the basic material on equivalence, discriminants, reduction, and class numbers, the author proceeds by comparing form class numbers and ideal class numbers, giving an alternative proof of the finiteness of the ideal class number, investigating the concept of ambiguity for forms and ideals, analyzing the genus of forms, and describing the assigned values of generic characters via Jacobi symbols. At the end of this chapter, the foregoing methods and results are applied to give numerous examples of representations of primes in the form \(p= a^2+ Db^2\), on the one hand, and to illustrate the equivalence of forms modulo a prime number \(p\) on the other.

Chapter 4 is devoted to the topic of Diophantine approximation, thereby assuming the elementary background on continued fractions, rational approximations, quadratic rationals, and related topics. Algebraic and transcendental numbers, the Thue-Siegel-Roth Theorem, Schanuel’s Conjecture, and a brief discussion of the famous number-theoretic constants of Gel’fond, Gel’fond-Schneider, Prouhet-Thue-Morse, Euler, Apéry, and Catalan are the main themes taken up here, along with a section on the relevant material from Minkowski’s geometry of numbers.

Chapter 5 provides some extended, more advanced knowledge of arithmetic functions, including Bernoulli numbers, Bernoulli polynomials, Fourier series, the Euler-Maclaurin summation formula, Wallis’s and Stirling’s formula, and the accurate approximation of the Euler-Mascheroni constant. Then, after a subsequent section on average orders of arithmetic functions and some concrete applications, the Riemann zeta function is briefly explained, in particular with a view toward its relations to the celebrated Prime Number Theorem and its various arithmetic function equivalences, on the one hand, and Riemann’s hypothesis on the other.

Chapter 6 turns to the methods of \(p\)-adic analysis in number theory, with the focus on Hensel’s Lemma, valuation theory for fields, Ostrowski’s Theorem, and the representation of \(p\)-adic numbers as power series.

The principal goal of Chapter 7 is to establish Dirichlet’s Theorem on prime numbers in arithmetic progressions. To this end, the author first discusses Dirichlet characters, Dirichlet \(L\)-functions, and the related conjectural “Generalized Riemann Hypothesis”. After a proof of Dirichlet’s Theorem along these lines, Dirichlet densities and some of their applications are exhibited, culminating in the theorem on Dirichlet density of primes in arithmetic progressions.

Chapter 8 gives several applications of the previous chapters to Diophantine equations. In the first part, Lucas-Lehmer functions and their various identities are used to analyze the solutions of the classical “generalized Ramanujan-Nagell equations” \(x^2- D= p^n\) and those of the Bachet equations \(y^2= x^3+ k\) thereby completing the discussion begun in Chapter 1. This is followed by a thorough treatment of Fermat’s Last Theorem for regular prime exponents, including Kummer’s pioneering proof of it, whereas the last section of this chapter is devoted to both the spectacular “ABC Conjecture” and the former “Catalan Conjecture”. The author depicts the present state of art regarding these milestones in number theory, explains the still unresolved “Fermat–Catalan Conjecture” in this context, and turns then to the crucial significance of the ABC Conjecture. Several consequences of the latter are derived, among them being the Thue-Siegel-Roth Theorem, Hall’s Conjecture, the Erdős-Mollin-Walsh Conjecture, the Granville-Langevin Conjecture, and other related (conjectural) statements.

Chapter 9 introduces elliptic curves over an arbitrary field, mainly with a view toward their arithmetic properties and applications. After displaying the basics, torsion points on elliptic curves, the Lutz-Nagell Theorem, Mazur’s Theorem, Siegel’s Theorem, and the reduction of rationals on elliptic curves in a survey-like manner, some recent applications are briefly described. These include H. W. Lenstra’s elliptic curve factoring method for odd composite natural numbers, his related elliptic curve primality test, and the effective primality proving algorithm by S. Goldwasser and J. Kilian [Proc. 18th STOC, 316–329 (1986)]. A brief section on elliptic curve cryptography, meant as a further application, concludes this chapter, with the focus on the Menezes-Okamoto-Vanstone elliptic curve cryptosystem (1991).

Chapter 10, the last chapter of the book, provides a concise introduction to modular forms, in general, and to elliptic modular functions in particular. This serves as the necessary background material for explaining the celebrated Shimura–Taniyama–Weil Conjecture both in terms of \(L\)-functions and modular representations. Using Ribet’s Theorem, it is demonstrated how this conjecture, which in 2001 has become what is now called the “Modularity Theorem”, implies the statement of Fermat’s Last Theorem.

Another central topic in number theory, namely sieve methods used to estimate the cardinalities of various sets of numbers defined by multiplicative properties, is treated in an appendix. Without proofs, the author provides here an overview of some classical open problems for which the use of sieve methods has led to remarkable recent advances. Along the way, the reader gets acquainted with Eratosthenes’s sieve in terms of the Möbius function, Brun’s constant, Selberg’s sieve, the Brun-Titchmarsh Theorem, Selberg’s sieve on twin primes, Selberg’s sieve on the Goldbach Conjecture, Artin’s Conjecture and the large sieve on it, the Bombieri-Vinogradov Theorem, the more recent Friedlander-Iwaniec Theorem, the Elliott-Halberstam Conjecture, and the nearly brand-new theorems by Goldston-Pintz-Yildirim on primes in tuples [cf. D. A. Goldston, J. Pintz and C. Y. Yildirim, The path to recent progress on small gaps between primes, Analytic number theory. A tribute to Gauss and Dirichlet. Proceedings of the Gauss-Dirichlet conference, Göttingen, Germany 2005. Providence, RI: American Mathematical Society (AMS). Clay Mathematics Proceedings 7, 129–139 (2007; Zbl 1213.11168)].

Finally, with these results as an illustration of the power of sieve methods, the use of the latter in tackling factorization problems is demonstrated by describing the powerful “Number Field Sieve Algorithm” due to Pollard, Buhler, H. W. Lenstra, Pomerance, and others.

In this context, the factoring of the ninth Fermat number is carried out as an instructive example.

As in the author’s foregoing, more elementary textbook “Fundamental Number Theory with Applications” [CRC Press Series on Discrete Mathematics and its Applications. Boca Raton, FL: CRC Press (1998; Zbl 0943.11001)], each section comes with a large number of illustrating examples and accompanying exercises. Solutions of the odd-numbered exercises are included at the end of the text, and a solutions manual for the even-numbered exercises is available to instructors who adopt the text for a course. Also, there are about 50 biographical sketches of mathematicians who contributed to the development of the theories depicted in the course of the text, and each section features an introductory quotation of a historical celebrity as a cultural appetizer. The rich bibliography contains 106 references, where maximum information is imparted by explicit page reference for each citing of a given item within the text. The carefully compiled index has more than 1,500 entries presented for maximum cross-referencing. Overall, this excellent textbook bespeaks the author’s outstanding expository mastery just as much as his mathematical erudition and elevated taste. Presenting a wide panorama of topics in advanced classical and contemporary number theory, and that in an utmost lucid and comprehensible style of writing, the author takes the reader to the forefront of research in the field, and on a truly exciting journey over and above.

Reviewer: Werner Kleinert (Berlin)

### MSC:

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11R04 | Algebraic numbers; rings of algebraic integers |

11E12 | Quadratic forms over global rings and fields |

11G05 | Elliptic curves over global fields |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11F03 | Modular and automorphic functions |