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B018797 - NUMBER THEORY
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Type of Assessment
Course program
Academic Year 2017-18
Coorte 2017 - Second Cycle Degree in MATHEMATICS
Course year
First year - First Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/02 - ALGEBRA
Credits
9
Teaching Hours
72
Teaching Term
18/09/2017 ⇒ 22/12/2017
Attendance required
No
Type of Evaluation
Final Grade
Course Content
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Course program
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Lectureship
Mutuality
Course teached as:
B018797 - TEORIA DEI NUMERI
Second Cycle Degree in MATHEMATICS
Curriculum DIDATTICO
B018797 - TEORIA DEI NUMERI
Second Cycle Degree in MATHEMATICS
Curriculum DIDATTICO
Teaching Language
Italian.
Course Content
The course can be considered to be divided in three parts.
The first resumes some preliminary notions about the theory of congruences. Moreover we prove Gauss quadratic reciprocity law. In the second section we study arithmetic functions and their averages. We give proofs of Bertrand's pustulate, and Dirichlet's Theorem. The third part is devoted to Additive Theory. Waring problem, functions g and G and The Schnirelmann's methods are treated.
The first resumes some preliminary notions about the theory of congruences. Moreover we prove Gauss quadratic reciprocity law. In the second section we study arithmetic functions and their averages. We give proofs of Bertrand's pustulate, and Dirichlet's Theorem. The third part is devoted to Additive Theory. Waring problem, functions g and G and The Schnirelmann's methods are treated.
Suggested readings (Search our library's catalogue)
1) My Lecture Notes.
Other suggested readings:
2) M. B. Nathanson, Elementary Methods in Number Theory.
3) M. B. Nathanson, Additive Number Theory.
3) T. Apostol, An Introduction to Number Theory.
Other suggested readings:
2) M. B. Nathanson, Elementary Methods in Number Theory.
3) M. B. Nathanson, Additive Number Theory.
3) T. Apostol, An Introduction to Number Theory.
Learning Objectives
The main goal of the course is to transmit knowledge of this fashinating theory. By the end of the course the students should be ready to read and understand the theorems shown at lessons and collected in the notes of the course. Moreover, they should apply this knowledge to solve exercises of diffferent difficulty.
Prerequisites
Preferably the full programs of the courses of Algebra 1 and 2 of the first two years of Laurea Triennale in Matematica.
Teaching Methods
Front lessons (rigorously given with blackboard and chalk). Moodle platform. Personal web page.
Type of Assessment
Oral exam.
Course program
The course is divided in three parts.
The first part resumes some preliminary notions
of first two years courses in Algebra, such as the greatest common divisor, Euclid algorithm and Bezout formula. Moreover we recall the theory of congruences and linear diofantine equations. We introduce important classes of prime and pseudoprime numbers (as Fermat and Mersenne primes and Carmichael numbers) and prove some primality tests (Lucas-Lehmer in primis). We also prove Hensel Lemma. The part ends with the theory of quadratic residues and a proof of the Gauss quadratic reciprocity law.
The second part is devoted to the Moltiplicative Theory. We study the arithmetic functions as elements of the commutative ring $C^{N^*}$ with usual sums and convolution product. We define and analyze the properties of the functions: d, sigma, Moebius mu and Euler phi function. We prove Moebius inversion formula and the caracterization of even perfect numbers (Euler Theorem). The asymptotic behaviour of the averages of these functions are determined. The course proceeds in the study of prime numbers and their distribution. We state the Prime Number Theorem and its consequences, we state and prove Čhebyshev Theorem and Bertrand Postulate. We briefly introduce the theory of caracters of abelian groups and prove Dirichlet Theorem on primes in progression.
The third part deals with the so-called Additive Theory. Waring problem and its variants are studied. We state and prove a Theorem of Lagrange and analyze the functions g and G. We introduce the concept of base for an additive structure and the Schnirelmann's method are treated, proving an important variation of the Goldbach conjecture. We conclude with a section devoted to partition and the proof of a formula of Ramanujan.
The first part resumes some preliminary notions
of first two years courses in Algebra, such as the greatest common divisor, Euclid algorithm and Bezout formula. Moreover we recall the theory of congruences and linear diofantine equations. We introduce important classes of prime and pseudoprime numbers (as Fermat and Mersenne primes and Carmichael numbers) and prove some primality tests (Lucas-Lehmer in primis). We also prove Hensel Lemma. The part ends with the theory of quadratic residues and a proof of the Gauss quadratic reciprocity law.
The second part is devoted to the Moltiplicative Theory. We study the arithmetic functions as elements of the commutative ring $C^{N^*}$ with usual sums and convolution product. We define and analyze the properties of the functions: d, sigma, Moebius mu and Euler phi function. We prove Moebius inversion formula and the caracterization of even perfect numbers (Euler Theorem). The asymptotic behaviour of the averages of these functions are determined. The course proceeds in the study of prime numbers and their distribution. We state the Prime Number Theorem and its consequences, we state and prove Čhebyshev Theorem and Bertrand Postulate. We briefly introduce the theory of caracters of abelian groups and prove Dirichlet Theorem on primes in progression.
The third part deals with the so-called Additive Theory. Waring problem and its variants are studied. We state and prove a Theorem of Lagrange and analyze the functions g and G. We introduce the concept of base for an additive structure and the Schnirelmann's method are treated, proving an important variation of the Goldbach conjecture. We conclude with a section devoted to partition and the proof of a formula of Ramanujan.