High School Mathematics Extensions/Primes/Definition Sheet

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 e &le; d.

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

, a solution exists if and only if for i and j with i &ne; 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.