**Definition**: Prime Numbers

A **prime number** or short a **prime** $p$ is a natural number, which has exactly two divisors, namely its trivial divisors \(1\) and \(p\).

We denote consecutive prime numbers by \(p_i, i=1,2,\ldots\) (for instance, \(p_1=2, p_2=3, p_3=5, p_4=7, p_5=11,\ldots\)) and the set of all primes as \[\mathbb P:=\{p_i,~i=1,2,\ldots\}.\]

| | | | | created: 2014-03-02 11:47:09 | modified: 2019-05-12 08:48:34 | by: *bookofproofs* | references: [1272]

## 1.**Proposition**: Natural Numbers and Products of Prime Numbers

## 2.**Definition**: Composite Number

## 3.**Proposition**: Existence of Prime Divisors

## 4.**Proposition**: Co-prime Primes

## 5.**Lemma**: Generalized Euclidean Lemma

## 6.**Proposition**: Greatest Common Divisors Of Integers and Prime Numbers

## 7.**Theorem**: Fundamental Theorem of Arithmetic

## 8.**Theorem**: Infinite Set of Prime Numbers

## 9.**Definition**: Perfect Square

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[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927

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