Math 255 Homework assignments

Homework Assignments

Math 255, Spring 2010

Professor Sands

Problems in parentheses are optional for undergraduate credit, required for graduate credit.
Graduate credit also calls for completing the following problems by the end of the semester: 30.2, 30.3, 39.1, 40.5
If you have questions, you can contact me using email: sands@cems.uvm.edu

# Due Date Homework Topic

1 Jan. 27
1.1, 1.2
2.1, 2.2 Number experiments
Pythagorean Triples

2 Feb. 3

29.1
3.1, 3.2 (3.3)
5.1a,5.3
Using algebraic numbers to solve diophantine equations
Rational points on unit circle
Greatest Common Divisors by Euclidean Algorithm

3 Feb. 10 6.2, 6.5 (6.4)
7.2, 7.3bc, 7.4, 7.5
Solve linear Diophantine equations by Euclidean Algorithm
Fundamental Theorem of Arithmetic

4 (Holiday Feb. 15)
Feb. 17 8.1, (8.2), 8.3, 8.4ce, 8.5
9.1, (9.2), (9.3), 9.4
Congruences
Fermat's Little Theorem

5 Feb. 24 10.2,10.3a,
11.1, 11.2a, (11.3), 11.5, 11.10, (11.13)
Euler-Fermat Theorem
Chinese Remainder Theorem and Euler Phi Function Property

6 (Test Mar. 3)
(Spring Break)
Mar. 17 12.2, 12.3, (12.5)
13.1, 13.5, (13,6)
14.1, (14.2), 14.3
Prime Numbers
Counting Primes
Mersenne Primes

7 Mar. 24 15.1,15.3 (15.4)
16.1 (16.3)
Perfect Numbers
Powers Modulo m

8 Mar. 31 17.2, (17.4), 17.5
18.1, (18.2) (Note 7081=73*97)
kth Roots Modulo m
RSA Cryptosystem

9 (Test April 5)
April 14 19.7a
Show gcd(a,n)=d iff gcd(a/d,n/d)=1
21.4, (21.5),
21.6, 21.9, (21.13), 22.1, 22.2, 22.4, (22.6) Rabin-Miller Primality Test
Euler's Phi-Function Summation Formula
Primitive Roots
Indices modulo p

10 April 21 23.1, 23.3a ,23.3d
24.1, (24.3), 24.4, 24.6
Squares modulo p
Are -1 and 2 squares mod p?

11 April 28 25.1, 25.3, 25.4, (25.6)
26.3
Quadratic Reciprocity
Which Primes are Sums of Two Squares?

12 May 5 27.1 (27.2) (27.6 Extra Credit)
28.1
Three problems from your assigned section.
For Grad. Credit: do problems listed at top.
Which Numbers are Sums of Two Squares?
Fermat's Last Theorem for Exponent 4

1	Jan. 27	1.1, 1.2 2.1, 2.2	Number experiments Pythagorean Triples
2	Feb. 3	29.1 3.1, 3.2 (3.3) 5.1a,5.3	Using algebraic numbers to solve diophantine equations Rational points on unit circle Greatest Common Divisors by Euclidean Algorithm
3	Feb. 10	6.2, 6.5 (6.4) 7.2, 7.3bc, 7.4, 7.5	Solve linear Diophantine equations by Euclidean Algorithm Fundamental Theorem of Arithmetic
4	(Holiday Feb. 15) Feb. 17	8.1, (8.2), 8.3, 8.4ce, 8.5 9.1, (9.2), (9.3), 9.4	Congruences Fermat's Little Theorem
5	Feb. 24	10.2,10.3a, 11.1, 11.2a, (11.3), 11.5, 11.10, (11.13)	Euler-Fermat Theorem Chinese Remainder Theorem and Euler Phi Function Property
6	(Test Mar. 3) (Spring Break) Mar. 17	12.2, 12.3, (12.5) 13.1, 13.5, (13,6) 14.1, (14.2), 14.3	Prime Numbers Counting Primes Mersenne Primes
7	Mar. 24	15.1,15.3 (15.4) 16.1 (16.3)	Perfect Numbers Powers Modulo m
8	Mar. 31	17.2, (17.4), 17.5 18.1, (18.2) (Note 7081=73*97)	kth Roots Modulo m RSA Cryptosystem
9	(Test April 5) April 14	19.7a Show gcd(a,n)=d iff gcd(a/d,n/d)=1 21.4, (21.5), 21.6, 21.9, (21.13), 22.1, 22.2, 22.4, (22.6)	Rabin-Miller Primality Test Euler's Phi-Function Summation Formula Primitive Roots Indices modulo p
10	April 21	23.1, 23.3a ,23.3d 24.1, (24.3), 24.4, 24.6	Squares modulo p Are -1 and 2 squares mod p?
11	April 28	25.1, 25.3, 25.4, (25.6) 26.3	Quadratic Reciprocity Which Primes are Sums of Two Squares?
12	May 5	27.1 (27.2) (27.6 Extra Credit) 28.1 Three problems from your assigned section. For Grad. Credit: do problems listed at top.	Which Numbers are Sums of Two Squares? Fermat's Last Theorem for Exponent 4