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Chapter: Elementary Results About Prime Numbers

In this chapter, we will introduce prime numbers and prove that some basic results about them, in particular, the generalized Euclidean lemma, the fundamental theorem of arithmetic, and that there are infinitely many prime numbers.

Table of Contents

1. Definition: Prime Numbers
2. Theorem: Infinite Set of Prime Numbers
3. Theorem: Fundamental Theorem of Arithmetic
4. Definition: Canonical Representation of Natural Numbers, Factorization
5. Definition: Subsets of Prime Numbers Not Dividing a Natural Number
6. Definition: Canonical Representation of Positive Rational Numbers
7. Definition: Floor and Ceiling Functions
8. Proposition: Number of Multiples of a Given Number Less Than Another Number
9. Proposition: Factorization of Greatest Common Divisor and Least Common Multiple

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References

Bibliography

1. Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
2. Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927