Arithmetic modulo m
The integers modulo m
From the CT we now know that the there are exactly m congruence classes modulo m: namely, m , m , . . . , [m
1]m . Lets give this set a name.
Denition. Let m be a modulus. The set of all congruence classes
The Prime Numbers
Before starting our study of primes, we record the following important
lemma. Recall that integers a, b are said to be relatively prime if gcd(a, b) =
Lemma (Euclids Lemma). If gcd(a, b) = 1 and a | bc then a | c.
Proof. This is an ap
What is the fastest way to compute a large integer power of a number modulo
For instance, suppose I want to compute 460 mod 69. One way to do this
is to just compute 460 in Z and then reduce the answer:
460 = 13292279957849158729
Linear Diophantine Equations
A diophantine equation is any equation in which the solutions are restricted
The word diophantine is derived from the name of the ancient Greek mathematician Diophantus, who was one of the rst people to consider s
In ordinary algebra, an equation of the form ax = b (where a and b are
given real numbers) is called a linear equation, and its solution x = b/a is
obtained by multiplying both sides of the equation by a1 = 1/a.
The subject of this lect
Functions and Cardinality of Sets
Real-valued functions of a real variable are familiar already from basic
(pre)calculus. Here we consider functions from a more general perspective,
in which variables are allowed to range over elements of arbitrary sets.
Fun with the Fundamental Theorem of Arithmetic
Denition. Given an integer with n = 0 and a prime p, the valuation of n at p, denoted vp (n), is the power
to which p is raised in the prime factorization of n. If p does no
Eulers Phi Function
An arithmetic function is any function dened on the set of positive integers.
Denition. An arithmetic function f is called multiplicative if f (mn) =
f (m)f (n) whenever m, n are relatively prime.
Theorem. If f is a multiplicative func
The Euclidean Algorithm
The Euclidean algorithm is one of the oldest known algorithms (it appears
in Euclids Elements) yet it is also one of the most important, even today.
Not only is it fundamental in mathematics, but it also has important applications
Division algorithm and base-b representation
An algorithm that was a theorem
Another application of the well-ordering property is the division algorithm.
Theorem (The Division Algorithm). Let a, b Z, with b > 0. There are unique i
Lets begin by recalling the denitions and a theorem. Let m be a given
modulus. Then the nite set
Z/mZ = cfw_, , . . . , [m 1] = cfw_0, 1, . . . , m 1
of residue classes modulo m is called the ring of integers modulo m. It
Everyone already knows certain divisibility tests. For instance, a number
(written in base-10 notation) is divisible by 10 i its last digit is a 0, divisible
by 100 i its last two digits are 00, etc. A number is divisible by 5 i its las
The congruence relation
The notion of congruence modulo m was invented by Karl Friedrich Gauss, and does much to simplify
arguments about divisibility.
Denition. Let a, b, m Z, with m > 0. We say that a is congruent to b modulo m, written
Lecture 1: Propositions and logical connectives
One of the stated objectives of the course is to teach students how to understand and fashion mathematical
arguments. Essential to and characteristic of these arguments is a precise logical structure. This r