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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 \[\mathbb P:=\{p_i,~i=1,2,\ldots\}.\]

Table of Contents

1. Proposition: Existence of Prime Divisors
2. Proposition: Natural Numbers and Products of Prime Numbers
3. Definition: Composite Number
4. Proposition: Co-prime Primes
5. Lemma: Generalized Euclidean Lemma
6. Proposition: Greatest Common Divisors Of Integers and Prime Numbers
7. Definition: Perfect Square

Mentioned in:

Chapters: 1
Corollaries: 2 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14
Examples: 15
Explanations: 16
Lemmas: 17 18 19
Parts: 20
Problems: 21
Proofs: 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Propositions: 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
Sections: 72 73 74 75
Solutions: 76
Theorems: 77 78 79 80 81 82 83 84

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References

Bibliography

1. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927