The sections referred to in the problems below come from the online text by Stein or Clark.

** HW #1:** Clark, 2.5, 2.6, 2.7, 2.9. 2.10. 2.11 (due 1/27/17).

** HW #2:** Clark, 7.3, 9.1, 9.2, 9.3. For 9.1, use the Euclidean algorithm and reverse substitution (due 1/27/17).

** HW #3:** Clark, 9.1, this time using Blankinship's method (due 1/27/17).

** HW #4:** Stein, 1.3, 1.8, 1.12, 1.13 (due 2/3/17).

** HW #5:** Stein, 1.7, 1.14 (due 2/3/17).

** HW #6:** Clark, 13.2, 13.3, 13.4 (due 2/10/17).

** HW #7:** #1. Let S denote the set of real numbers. For a, b in S, define a ~ b to mean a-b is in **Z**. Prove that
~ is an equivalence relation.

#2. Let T denote the ordered pairs of integers (a, b), with b non-zero.
For (a, b) and (c, d) in T,
define (a, b) ~ (c, d) to mean ad=bc. Prove:

- ~ is an equivalence relation.
- If (a, b) ~ (a', b') and (c, d) ~ (c', d'), then (ad+bc, bd) ~ (a'd'+b'c', b'd').
- If (a, b) ~ (a', b') and (c,d) ~ (c', d'), then (ac, bd) ~ (a'c', b'd').

** HW #8:** Clark, 15.4, 15.5, 15.6, 15.7, 15.8 (due 2/17/17).

** HW #9:** #1. Write out addition and multiplication tables for **Z**_{4} and **Z**_{12}.
#2. Consider the equations 3x ≡ 2 and 2x ≡ 10. Use #1 to find all solutions to these equivalences (separately) in
both **Z**_{4} and **Z**_{12}, (due 2/17/17).

** HW #10:** Due 2/24/17.

- Prove that for positive integers b,c and a prime number p, (b+c)
^{p}≡ b^{p}+ c^{p}(mod p). Hint: Use the binomial theorem. - Find a solution to the system of congruences: x ≡ 4 (mod 6), x ≡ 6 (mod 7), and x ≡ 10 (mod 13).
- Show that your answer in #2 is unique, modulo 546.

** HW #11:** Let φ(n) denote the Euler phi function. Use the formulas from class to derive the following
(due 2/24/17).

- φ(ab) = φ(a)φ(b) · (d ÷ φ(d)), where d = GCD(a,b).
- If a|b, then φ(a) | φ(b).

** HW #12:** Stein, 2.8, 2.13, 2.23 (due 3/3/17).

** HW #13:** Stein, 2.5, 2.10, 2.11, 2.14. Note for 2.11, you are being asked to find the least positive r for
which 2^{r} is congruent to 1 modulo 17 (due 3/3/17).

** HW #14:** Stein, 2.24, 2.26, 2.27(a) (due 3/10/17).

** HW #15:** Stein 2.33 and the following two problems, for which p denotes a prime number (due 3/10/17).

- Let f(x) = a
_{d}x^{d}+ · · · + a_{0}and g(x) = b_{d}x^{d}+ · · · + b_{0}belong to**Z**[x]. Suppose a_{i}≡ b_{i}(mod p), for all i. Prove that an integer u is a root of f(x) modulo p if and only if u is a root of g(x) modulo p. - Let f(x) = ax
^{2}+bx+c belong to**Z**[x] and assume a is not divisible by 2. Find all roots of f(x) modulo 2. Hint: Use the previous problem.

** HW #16:** Stein, 4.1, 4.8 and the following: Use Euler's criterion to find the quadratic residues modulo 19 and then
find both square roots of these residues modulo 19 using the fact that 19 ≡ 3 (mod 4). (due 3/17/17)

** HW #17:** In the problems below, (a/p) denotes the Legendre symbol of a over p (due 3/17/17).

- Use the law of quadratic reciprocity and properties of the Legendre symbol to calculate (-42/61), (-1/61), (2/61), (7/61).
- Use the law of quadratic reciprocity to determine if 3x
^{2}+10x-11 has a root modulo 227. You need not find a root, if a root exists modulo 227. - Find the prime numbers p less than 50 for which (5/p) = 1.

** HW #18:** Use Gauss's Lemma to calcluate: (2/17), (3/11) and (9/13). All details must be provided. (due 3/31/17)

** HW #19:** Stein 4.3, and the following: Use Gauss's Lemma to prove (2/p) = -1, if p ≡ 5 (mod 8). (due 3/31/17)

** HW #20:** For the two pairs of Guassian integers, (2-12i, 6+8i) and (24-i, 36+24i), find a GCD for each pair, and express
that GCD as a linear combination of the pair, as guaranteed by Bezout's Principle. (due 4/21/17)

** HW #21:** Factor each of the following Gaussian integers as a product of Gaussian primes: 7350, 3+6i, 1+5i. (due 4/21/17)

** HW #22:** For Gaussian integers x= 6+8i, y = 3+11i, and z= 1+2i find Gaussian integers x', y' such that x ≡ x' (mod z) and y ≡ y' (mod z), with N(x') < N(z) and N(y') < N(z).
Then verify x+y ≡ x'+y' (mod z) and xy ≡ x'y' (mod z). (due 4/28/17)

** HW #23:** For the Gaussian priime 2+i, find a distinct set of residue classes modulo 2+i, and for the
four non-zero classes, find the order of each class modulo 2+i. (due 4/28/17)

** HW #24:** Solve the following congruences over the Gassian integers: (a) (3+5i)x ≡ 7-i (mod 3-5i) and
(b) (1+i)x ≡ 10-3i (mod 3+2i). (due 5/3/17)

** HW #25:** For the Gaussian integers z = 2+3i and y = 3-i,
find a full set of distinct residue classes, so that the norm of each representative of the
class is less than the norm of the Gaussian integer defining the class. (due 5/3/17)