Math 453: Denitions and Theorems
4
3-18-2011
A.J. Hildebrand
Quadratic residues
4.1
Quadratic residues and nonresidues
Denition 4.1 (Quadratic residues and nonresidues). Let m N and a Z be such that
(a, m) = 1. Then a called a quadratic residue modulo m i
Math 453, Section X13
Midterm Exam 2 Solutions
Spring 2011
Problem 1
For this problem just state the requested results; no proofs required.
(i) State the Moebius Inversion Formula. Be sure to include any assumptions and any necessary quantiers
(e.g., for
Math 453: Elementary Number Theory
Denitions and Theorems
(Class Notes, Spring 2011 A.J. Hildebrand)
Version 1-30-2011
1
Math 453 Denitions and Theorems
1-30-2011
A.J. Hildebrand
About these notes
One purpose of these notes is to serve as a handy referenc
Math 453: Denitions and Theorems
3
3-7-2011
A.J. Hildebrand
Arithmetic functions
3.1
Some notational conventions
Divisor sums and products:
Let n N.
f (d) denotes a sum of f (d), taken over all positive divisors d of n.
d|n
f (p) denotes a sum of f (p), t
Math 453: Denitions and Theorems
2
2-16-2011
A.J. Hildebrand
Congruences
2.1
Denitions and basic properties; applications
Denition 2.1 (Congruences). Let a, b Z and m N. We say that a is congruent to b
modulo m, and write a b mod m, if m | a b (or, equiva
Math 453: Denitions and Theorems
6
6.1
4-18-2011
A.J. Hildebrand
Continued fractions
Denitions and notations
Denition 6.1 (Continued fractions). A nite or innite expression of the form
(6.1)
1
a0 +
1
a2 + . . .
a1 +
,
where the ai are real numbers, with a
Name:
Collaborator(s)1 :
Math 453, Section X13, Prof. Hildebrand, Spring 2011
HW Assignment 8, due Friday, 4/8/2011
Instructions
Rules: The usual: Write your name on the cover sheet and staple the sheet to the assignment. The
assignment is due in class a
Math 453: Denitions and Theorems
7
5-4-2011
A.J. Hildebrand
Topics in Computational Number Theory
7.1
Some Basic Concepts
Running Time: A fundamental notion that allows one to quantify and compare the ecieny
of algorithms. The running time of an algorithm
Math 453: Denitions and Theorems
5
3-30-2011
A.J. Hildebrand
Primitive roots
5.1
The order of an integer
Denition 5.1 (Order of an integer). Let m N and a Z be such that (a, m) = 1. The order
of a modulo m, denoted by ordm a, is the least positive integer
Name:
Collaborator(s)1 :
Math 453, Section X13, Prof. Hildebrand, Spring 2011
HW Assignment 4, due Monday, 2/21/2011
Instructions
Rules: The usual: Write your name on the cover sheet and staple the sheet to the assignment. Do the
problems in order, and m
Math 453, Section X13
Exam 1 Solutions
Spring 2011
Problem 1
True/false questions. For each of the following statements, determine if it is true or false, and provide a
brief justication for your claim. Credit on these questions is based on your justicati
Math 453, Section X13
Final Exam Solutions
Spring 2011
Problem 1
True/false questions. For each of the following statements, determine if it is true or false, and provide
a brief justication for your claim. Credit on these questions is based on your justi