### Math 575: Introduction to Number Theory: Syllabus

Text (required): I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, John Wiley & Sons, 1991.

This syllabus is tentative. It will be superseded by later versions as the course evolves. (Oct. 17, 2009 version)

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 # date sections in Niven and Zuckerman HW due dates problems 1 9/9 History: Babylonian/Greek 2 9/11 History: through Gauss, Sec. 1.2 induction, start theorem 1.2 3 9/14 Sec. 1.2 GCD; state Euclidean algorithm 4 9/16 Sec.1.3 GCD, LCM, start fundamental theorem of arithmetic 5 9/18 Sec. 1.3 Fundamental theorem of arithmetic; infinity of primes, and primes of form 4k+3, \sum 1/p > loglog x -C HW#1 due Fri. 9/18 6 9/21 Sec. 1.4 Binomial coefficients, Sec. 2.1 congruences 7 9/23 Facts on primes, Sect. 2.1, congruences, Euler's Thm. [ Lecture: DeBacker] 8 9/25 Fermat's little thm (Euler thm) Sec. 2.1 (Wilson's thm.), 2.2 (linear congruences), 2.3 (Chinese remainder theorem) [Lecture: Prasanna] 9 9/28 Sec. 2.4 computing powers (mod m), 2.3 Chinese remainder theorem =Lagrange interpolation, Sec. 2.1 (sum of two squares), HW#2 due Tues. 9/29 10 9/30 Sec. 2.3 (multiplicative functions) Sect. 4.2 (Theorem 4.4) multiplicative function 11 10/2 Sec 2.5 Public key cryptography, computing exponents 12 10/5 Facts on factoring and primality testing, Sect. 2.7 congruences (mod p), Lagrange's theorem 13 10/7 Sect. 2.7 Factoring x^n-1, Wolstenholme's theorem Sect. 2.6 prime power moduli: Taylor expansion, binomial theorem, Hensel's lemma, nonsingular case 14 10/9 Sect. 2.6 Hensel's lemma: singular case; p-adic numbers, absolute values HW#3 due Fri. Oct. 9 15 10/12 p-adic numbers-continued; Ostrowski theorem statement, 16 10/14 Sect. 2.8 Primitive roots (mod p). 17 10/16 Midterm Exam 1 (in class) Covers Chapters 1 and 2 through 2.7: 10/19 Fall Study Break 18 10/21 Sect. 2.8 Primitive roots (general case); quadratic residues primality testing 19 10/23 Sect. 2.9 Applications of primitive roots; power residues 20 10/26 3.1 Quadratic Residues, Gauss Lemma , HW#4 due Mon. Oct. 26 21 10/28 3.2 quadratic reciprocity 22 10/30 3.2/3.3 Quadratic reciprocity, Jacobi symbol 23 11/2 3.3 Counting solutions to two-varible congruences 23 11/4 3.4 Binary quadratic forms HW#5 due Wed. Nov. 4 24 11/6 3.5 Equivalence of binary quadratic forms; class number 25 11/9 3.5 Reduction of binary forms 26 11/11 3.7 Positive definite forms,examples 27 11/13 3.7 Positive definite forms, unique reduction 28 11/16 3.7 Positive def. forms, automorphs, representations HW#6 due Mon. Nov. 16 29 11/18 4.2 Arithmetic Functions 30 11/20 4.2, 4.3 Arithmetic functions/ Mobius inversion 31 11/23 Midterm 2 (Mon. Nov. 23) Covers Chapter 2.8: Chap. 3 [Does not cover Chap. 4] 32 11/25 8.3 Estimates of arithmetical functions, d(n) 11/27 Thanksgiving 33 11/30 8.3 Estimates of Arithmetical functions, squarefree, sums of two squares 34 12/2 7.1-7.2 Diophantine approximation, continued fractions HW#7 due 35 12/4 7.1, 7.2 Continued fractions 36 12/7 7.7 Periodic continued fractions 37 12/9 7.7, 7.8 Diophantine equations; Pell's equation HW#8 due Take-home Final Exam handed out : 38 12/11 Liouville's theorem; irrationality 39 12/14 Diophantine equations; even and odd Perfect numbers Take Home Final Due 5pm