Math 118: Advanced Number Theory
Samit Dasgupta and Gary Kirby
April 1, 2015
Contents
1 Basics of Number Theory
2
1.1 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2 The Euclidean Algorithm and Unique Factorizat
MATH 118 HW 7
KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN
LASKE
Part 1. Let p be an odd prime and let a 2 Z such that gcd(a, p) = 1.
Show that if a is a quadratic residue mod p, then a is a quadratic
residue mod pn for any positive integer n.
Pro
Homework #35
Homework #
Jackson Hsu
Math 118
April 28, 2015
Recall the integral domain
R=O
19
=
p
a+b
19
: a, b 2 Z, a b
2
(mod 2)
= Z[],
p
where = (1 +
19)/2. The goal of this assignment is to prove that R is a PID but not
a Euclidean Domain.
Part 1: Pro
MATH 118 PROBLEM SET 6
WASEEM LUTFI, GABRIEL MATSON, AND AMY PIRCHER
Section 3.1
#16: Show that if a is a quadratic residue modulo m, and ab 1 (mod m), then b
is also a quadratic residue. Then prove that the product of the quadratic residues
modulo p is c
MATH 118
HOMEWORK #5
BRANDEN LASKE, TANYA PARAON, AND AMY PIRCHER
Section 2.8
#9: Show that 38 1 (mod 17). Explain why this implies that 3 is a primitive root
of 17.
Proof. First we calculate 34 = 81 4 (mod 17), so 38 = (34 )2 (4)2 16 1
(mod 17).
Now reca
Math 118 Homework #2 Solutions
April 23, 2015
Section 1.3, #53.
Let (x) = k, and let the primes less than or equal to x be p1 = 2, p2 = 3, . . . , pk , in
increasing order. The function (u) is a step function that is equal to 1 for p1 u < p2 ,
equal to 2
HOMEWORK # 4
MARIA SIMBIRSKY
SANDY ROGERS
MATTHEW WELSH
1. Section 2.1, Problems 5, 8, 28, and 48
Problem. (2.1.5) Write a single congruence that is equivalent to the pair of congruences
x 1 (mod 4) and x 2 (mod 3).
Solution. x 1 (mod 4) and x 2 (mod 3) i
Homework #8 Solutions
Math 118, Winter 2015
June 9, 2015
1. A loose end from class: prove that if Q(x, y) = Ax2 + Bxy + Cy 2 is a quadratic form, then
the constant M = |A| + |B| + |C| satises the property that |Q(a, b)| M max(|a|2 , |b|2 ) for
all real nu
MATH 118 HW 1
LINDSAY CROSS, KELLY DOUGAN, KEVIN LIU
1.2
Problem 2. Find the greatest common divisor g of the numbers 1819
and 3574, and then nd integers x and y to satisfy 1819x + 3587y = g.
Use the Euclidean Algorithm to nd gcd(3587, 1819):
3587 = 1819(