## Introduction to Number Theory, Math 104, Spring quarter 2000:

Course material: We use the book `Number theory with computer applications' by Kumanduri and Romero. No programming will be required. A tentative outline of the material to be covered is given below. This may change during the course, or we may move faster or slower than projected:

First quarter: chapters 1-5, 7, 9,

Second quarter: chapters 11, 14, 17, 19 possibly more material on elliptic curves, possibly chapter 15,

Office hours: MF: 1-2 and by appointment

Office: APM 5256, tel. 534-2734

Teaching assistant: Eric Rowell, email: rowell@euclid.ucsd.edu, office hours:

Computation of grade: The grade is computed from your scores in the final (50 %), 1 midterm (30%), the best 2 of 3 quizzes (15%) and homework (5%) (best 4 out of 5).

Dates of quizzes and midterm:

quizzes: 4/19, 5/3, 5/31 (wednesdays)

midterm: 5/17

Homework assignments: (in weeks without a quiz or midterm you will have to turn in selected homework problems in the TA SECTION which will be graded; the problem(s) to be turned in will be announced in section).

Disclaimer: I will try to get the homework assignment on the net in time. Due to time and other limitations, this may not always be possible. The fact that there is no assignment posted for a particular date does therefore NOT necessarily mean that no homework is due.

for 4/11: p.248: 1ab, p.255: 1bc, 3, 6, 11, 16

for 4/18 (quiz on 4/19): p. 262: 1, 3, 8, 11, p. 268: 2, 3, 4, (if you have difficulties with 2,3,4 let me know on Monday)

for 4/25: p330: 2, 3, 7, 12, p352: 1, 5

for 5/2 (quiz on 5/3): p486: 2, 3ab, 4ab, p497: 1, 4, 5 (WARNING: the formula for the discriminant in the book is WRONG: the polynomial is f(x)=x^3 + ax^2 + bx + c, and in the formula you have to replace the last term by -18abc)

for 5/9: p498: 9, p503: 3,4 and the following

Problem: Consider the elliptic curve E: y^2=x^3+17. a) Compute the number N_p of solutions of E mod p for p=7,13, 19 (don't forget the point O at infinity)! b) Let a_p=p+1-N_p. Compute the quantity 4p-a_p^2 for p as in a), and also for p=31(a_p=-11) and for p=37(a_p=-11). Can you guess a general pattern? (sorry, the values for a_31 and a_37 I gave in class were accidentally taken from another example, the values here are the correct ones)

for 5/16: p508: 5,7,

Midterm on 5/17: material will be from the beginning of the quarter until th assignment for 5/16. The usual rules apply: one cheat sheet is allowed, as well as calculators; no books or other notes. review exercises: p 347: 9, 11 (see Prop. 14.5.3, Theorem 14.5.5)

for 5/23: p221: 2ab, 4, 7, 13, p229: 2

for 5/30 (quiz on 5/31): p230: 6abc, p420: 1abf, 2, 3

for 6/6: p431: 4, 5ab, 8, p435: 1ab, 2