HW 0
Problem 1. We know from the Proposition on page 5 of Hatcher that each choice
of 1 < p gives us a triple of the form
(2pq, p2 q 2 , p2 + q 2 ).
It is also easy to verify that each triple comes from one and only one (p, q ). See
Hatcher or try it your
PRACTICE PROBLEMS:
PRIMITIVE ROOTS AND ARITHMETIC FUNCTIONS
Problem 1.
(a) Find a primitive root mod 13.
(b) Find all primitive roots mod 13.
Solution.
(a) We need to nd a Z with order (13) = 12, so it suces to nd a Z coprime to 13
with a4 1 mod 13 and a6
Quiz I: Solutions
Math 248
November 19, 2012
1. (a) Show that 3 is a primitive root mod 17. Hint : 27 10 mod 17 and 100 2 mod 17.
Solution. (17) = 16, so we just need to show that 38 1 mod 17. 38 = 32 (33 )2
9 102 9 (2) 1.
(b) Find all residues mod 17 of
LECTURE NOTES: QUADRATIC FORMS
BENJAMIN BAKKER
These notes are a basic outline of what we proved in lecture about quadratic forms; Jones & Jones does
not cover this material, though it is treated in Davenport and some of the other references given in the
PRACTICE PROBLEMS:
PRIMITIVE ROOTS AND ARITHMETIC FUNCTIONS
Problem 1.
(a) Find a primitive root mod 13.
(b) Find all primitive roots mod 13.
Problem 2. What is the order of 6 mod 19?
Problem 3. Find a primitive root mod 11100 .
Problem 4. Suppose we have
Midterm I
Math 248
October 11, 2012
Name:
1
1. (10 points) Let p N be a prime. The order mod p of an integer x Z, x 0 mod p, is
the smallest number d N such that xd 1 mod pit is denoted ordp (x). By Fermats
little theorem, the order exists.
(a) Show that
Midterm I: Solutions
Math 248
October 11, 2012
1. (10 points) Let p N be a prime. The order mod p of an integer x Z, x 0 mod p, is
the smallest number d N such that xd 1 mod pit is denoted ordp (x). By Fermats
little theorem, the order exists.
(a) Show th
LECTURE NOTES 9/19
BENJAMIN BAKKER
The goal of these notes is to generalize our proof of the Fundamental Theorem of Arithmetic to other
rings. Before we understand what that means, consider the following set:
Denition 1. Z[i] = cfw_a + bi|a, b Z C is the
HW 2
Problem 1. The proof is identical to Theorem 2.9 in Jones. Primes can only
be of the form 1 mod 6 or 5 mod 6, since otherwise theyre divisible by 2, 3 or 6.
Suppose there are only a nite number of primes of the form 6x + 5; let them be
p1 , p2 , . .
HW 1
Problem 1. Let S N be a nonempty bounded set and let G be the set of upper
bounds of S . Since S has at least one upper bound, G is non-empty. By the wellordering principle, G has a least element g . We only have to check that g S .
Suppose not; sinc
Chapter 0
Preview
1
Chapter 0: A Preview
Pythagorean Triples
As an introduction to the sorts of questions that we will be studying, let us consider right triangles whose sides all have integer lengths. The most familiar example
is the (3, 4, 5) right tria