I. Niven, H. Zuckerman, H. Montgomery,An Introduction to the Theory of Numbers, 5th edition.

Midterm I: Monday, Mar 2, in class.

Midterm II: Monday, Apr 20, in class.

The second exam will cover the material learned since the first exam: polynomial congruences, primitive roots and power residues, quadratic residues and reciprocity, Legendre/Jacobi symbols, sums of squares, Pythagorean triples, and some ad hoc methods of solving equations in integers (sec. 5.4.)

Important: Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. You are welcome to collaborate with others and to consult books, but your solutions should be written up in your own words, and all your collaborators and sources should be listed.

Homework 1 (pdf), due Feb 4. Some solutions

Homework 2 (pdf), due Feb 11. Some solutions

Homework 3 (pdf), due Feb 18. Some solutions

Homework 4 (pdf), due Feb 25. Solutions

Homework 5 (pdf), due Mar 4.

Homework 6, due Mar 18: 4,7 sec. 2.6; 3 sec. 2.7; 2, 9 sec. 2.8. Solutions

Homework 7, due Mar 25: 8, 10, 18, 20, 22, 37 sec. 2.8. Hint for question 37: consider two cases, n even and n odd. For n odd, let p be the least prime divisor of n, and show that (p-1,n)=1. Then think about the order of 2 modulo p. Solutions

Homework 8, due Apr 15: 2, 9, 10 sec. 3.2; 1, 2 sec. 3.3; 8 sec. 5.3, and one more question: find all integers that can be represented as a difference of two squares, i.e. n=a²-b² . This is a lot easier than sums of squares! Solutions

Homework 9 (pdf), due Apr 29. There's a typo on the problem sheet, I meant to assign qiestion 3 from section 7.3 not 7.2. Solutions

Homework 10, due Mar 25: 1, 2, 3 sec. 7.4; 3, 4 sec. 7.5; 1 sec. 7.6. Some solutions

• Divisibility, Euclidean algorithm,
• Prime numbers, Fundamental theorem of arithmetic
• Linear Diophantine equations / Congruences
• Fermat's Little Theorem, Euler's Formula
• Powers modulo m
• Public key cryptography
• Primitive roots
• Quadratic residues
• Sums of squares
• Farey fractions, rational approximation, continued fractions
• Elliptic curves

Students with Disabilities: If you have a physical, psychological, medical, or learning disability that may impact on your ability to carry out assigned course work, you are strongly urged to contact the staff in the Disabled Student Services (DSS) office: Room 133 in the Humanities Building; 632-6748v/TDD. The DSS office will review your concerns and determine, with you, what accommodations are necessary and appropriate. A written DSS recommendation should be brought to your lecturer who will make a decision on what special arrangements will be made. All information and documentation of disability is confidential. Arrangements should be made early in the semester so that your needs can be accommodated.