HOMEWORK ASSIGNMENT 1  (due on Monday, January 24th)

Section 1.2:   2, 14, 23

Section 1.3:   5, 12

Section 2.1:   6, 10, 11, 20, 21, 23, 28, 29, 49

Section 2.2:  5d, 5f, 8

A: Find all the remainders that the square of an integer can give when divided 7.

B: Find all the remainders that the cube of an integer can give when divided 7.

C: Find all the remainders that the cube of an integer can give when divided 5.

D: Find the last digit in the decimal representation of 2²¹⁶⁰⁹¹ - 1.

E: (a) Let a and b be integers, not both zero. Let d = (a, b). Prove the following
statement: If c | a and c | b, then c | d.

(b) Prove Divisibility Proposition 11. (Note: A rather direct proof can be
given using the Division Algorithm.)

HOMEWORK ASSIGNMENT 2   (due Friday, February 4th)

Section 2.3:  1, 2, 4, 10, 13, 16, 17, 18

A. Find all solutions to the following congruence: x² + 1  =  0 (mod 65).

B. For each of the integers a = -1, 1, 2, and 5, compute ord₄₇(a). Compute ord₁₇(a).

C. Find all integers a such that ord₇(a) = 3. Find all integers a such that ord₁₁(a) = 3.

D. Find all primes p such that ord_p(2) = 11. Find all primes p such that ord_p(2) = 12.

E. Find the last two digits in the base 10 representation of 2²¹⁶⁰⁹¹ - 1. (Hint: Euler's theorem for m = 25 will be helpful in this problem.)

HOMEWORK ASSIGNMENT 3   (due Monday, February 28th)

Section 2.8:  3, 10, 14

Section 2.9:  1

Section 3.1:   1, 3, 4, 8, 9, 19

A. Let m = 2821. Assume that (a, m) = 1. Using the result from Exam Question 1, part (b), deduce that ord_m(a) divides 60. Find an integer a such that ord_m(a) = 60.

B. Suppose that p is a prime and that p = 3 (mod 4). Assume that p doesn't divide a. Prove that a is a quadratic residue modulo p if and only if ord_p(a) is odd.

C. Suppose that p is a prime and that p > 3. We have proved in class that the congruence x² = -1 (mod p) is solvable if and only if p = 1 (mod 4). We have also proved that the congruence x² = -3 (mod p) is solvable if and only if p = 1 (mod 3). Using these results, prove that the congruence x² = 3 (mod p) is solvable if and only if p = 1 or 11 (mod 12).

D. Let p = 257. Suppose that p doesn't divide a. Prove that a is a primitive root modulo p if and only if a is a quadratic nonresidue modulo p.

HOMEWORK ASSIGNMENT 4 (due Monday, March 7th)

Section 3.1: 7a, b, c, d

Section 3.2: 1, 2, 5, 6, 7

A. Suppose that p is a prime and that p = 1 (mod 4). Let k = (p-1)/4. Prove that there are exactly k quadratic nonresidues b in the interval 0 < b < p/2.

B. Is the congruence x² = 222 (mod 257) solvable?

C. For which primes p is the congruence x² = -7 (mod p) solvable?

D. Suppose that p is a prime with the property that q = 2p + 1 is also a prime. Assume also that p = 3 (mod 4). Prove that q divides 2^p - 1.