High School Mathematics Extensions/Primes/Definition Sheet

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High School Mathematics Extensions

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Supplementary Chapters — Primes and Modular Arithmetic — Logic

Mathematical Proofs — Set Theory and Infinite Processes — Counting and Generating Functions — Discrete Probability

Matrices — Further Modular Arithmetic — Mathematical Programming — Markov Chains

About the definition sheet:

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.

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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 ≤ 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.

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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(m_i,m_j) divides (a_i - a_j)

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.