High School Mathematics Extensions/Primes/Definition Sheet

Content HSME Primes Modular Arithmetic Problem Set Project Exercise Solutions Problem Set Solutions Definition Sheet Full Version PDF Version

Contents

The definitions provided below may differ a little from the ones given in the chapter. If possible, print out the following definitions for easy reference.

Definitions

Composite

A composite is a whole number that is not a prime. The number 1 is not composite.

Coprimes

Two numbers are coprimes if their greatest common divisor (gcd) equals 1.

Diophantine Equation (linear)

An equation of the form ax + by = c. Where a, b and c are integer constants, and x and y are unknown integers.

Factorisation

Alternatively spelt factorization. A process in which the prime factors of a natural number are found and the number expressed as the product of the individual factors.

gcd (greatest common divisor)

The gcd of a and b is a number d, such that d divides a and d divides b; and that if e divides a and b, then ed.

Inverse

In modular m arithmetic, the inverse of a is the number b such that
$ab \equiv 1 \pmod{m} \!$
the inverse is unique. Not every number in every arithmetic have an inverse.

Modular arithmetic

The arithmetic modulo m is the arithmetic where each number is represent by a number lying between 0 and m - 1. E.g. consider modulo 7 arithmetic, 11 is represented by 4; and -2 is represented by 5. We say 11 is equivalent to 4 mod 7; and -2 is equivalent to 5 mod 7. It is explained in more detail here.

Prime

A prime number (or prime for short) is a whole number that can only be wholly divided by two different numbers, 1 and itself. The number 1 is thus not considered prime. We do not consider the negative numbers in this chapter.

Theorems

Chinese Remainder Theoren

In a system of n congruencies

$x \equiv a_1 \pmod{ m_1}\,\!$
$x \equiv a_2 \pmod{ m_2}\,\!$
...
$x \equiv a_n \pmod{ m_n}\,\!$

, a solution exists if and only if for i and j with i ≠ j

gcd(mi,mj) divides (ai - aj)

Existence of inverse

In modular m arithmetic, a has an inverse if and only if gcd(a,m) = 1.

Fundamental Theorem of Arithmetic

Any integer (except for 1) can be expressed as the product of primes in one and only one way.

Infinitely many primes

There are infinitely many primes.