MATH 310: Homework 2
Due Tuesday, 9/20 in class
Reading: Davenport I.4, I.5
1. Show that xn y n = (x y)(xn1 + xn2 y + + xy n2 + y n2 ).
2. Show that if 2n 1 is prime, then n is prime. Is the converse true?
3. Show that for n odd, xn + y n = (x + y)(xn1 xn
1
MATH 310: Homework 3
Due Tuesday, 9/27 in class
Reading: Notes on G.C.D.; Davenport p.15, I.6, I.7
1. Show that d|m and d|n if and only if d| gcd(m, n).
2. Prove the formula
lcm(m, n) =
Y
pmax(ordp m,ordp n) .
p
3. Show that if p, q, r are relatively pr
MATH 310: Homework 1
Due Thursday, 9/8 in class
Reading: Davenport I.1, I.2, I.3
1. Explain how the second law of cancellation (on page 2, line -1)
ax = ay
implies
x=y
follows from the properties of <.
2. Explain why the solution to b + x = a, if it exist
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MATH 310: Homework 4
Due Thursday, 10/6 in class
Reading: Davenport I.8, I.10
1. Find one solution to the Diophantine equation 113x 355y = 1.
2. Describe all solutions to the Diophantine equation 113x 355y = 1.
3. Show that ax + by = n has a solution in
1
Notes on Greatest Common Divisor
Let N = cfw_1, 2, 3, . . . denote the set of natural numbers and N0 = N cfw_0.
Definition 1. The largest exponent e such that pe divides n is denoted by
ordp n := maxcfw_e N0 : pe |n.
Theorem 1. Let p be a prime that oc
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MATH 310: Homework 7
Due Thursday, 12/1 in class
Reading: Davenport III.1, III.2, III.3, III.4, III.5
1. Show that x is a root of unity modulo m if and only if (a, m) = 1.
2. Find all primitive roots mod 11.
3. Find all primitive roots mod 27.
4. Find a