# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 115B: Number Theory

**Approved:**2003-05-01 (revised 2013-01-01, G. Kuperberg)

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

Search by ISBN on Amazon: 978-0321500311

**Prerequisites:**

**Suggested Schedule:**

Lecture(s) |
Sections |
Comments/Topics |

1-5 |
ch. 7.4, 7.3 |
Famous functions of number theory: Möbius inversion, multiplicative functions, sum of positive divisors, Morton’s conjecture, perfect numbers |

6-13 |
ch. 9.1-9.3 ch. 10.1, ch. 10.2 |
More congruences, primitive roots, applications: pseudorandom numbers, the ElGamal Crypto System |

14-21 |
ch 11.1-11.3 |
Quadratic reciprocity, Legendre symbol, Jacobi symbol, the law of quadratic reciprocity |

22-27 |
ch. 13 |
Nonlinear Diophantine, equation and continued fractions, Pythagorean triples, Fermat’s last Theorem, Pell’s equation, sums of squares |

**Additional Notes:**

**Learning Goals:**

Mastery of this course enhances the students' ability to construct and write proofs; to not only see beautiful ideas of number theory in the time of Gauss, but also reach some of those ideas themselves; and adds to their experience with algebra in general, in particular in association with the Math 150 modern algebra series. AS A CAPSTONE: Students will develop and deepen their understanding of number theory in this second course of the MAT 115 sequence. Through applications to cryptography they will further their understanding of the role number theory plays in the modern world. Students will gain a deeper appreciation of the unity of mathematics by seeing the many different mathematical tools employed to solve number theoretic problems. Via this subject area expertise, they will gain mastery in this area of specialization and improve their ability to communicate mathematics at a capstone level, commensurate with that expected of one with an undergraduate degree in mathematics.

**Assessment:**