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The Number of Primes Below a Given Limit

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Prime Numbers and Computer Methods for Factorization

Part of the book series: Modern Birkhäuser Classics ((MBC))

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Abstract

Consider the positive integers 1,2, 3,4, … Among them there are composite numbers and primes. A composite number is a product of several factors ≠ 1, such as 15 = 3∙5; or 16 = 2∙8. A prime p is characterized by the fact that its only possible factorization apart from the order of the factors is p = 1 p. Every composite number can be written as a product of primes, such as 16 = 2∙2∙2∙2.— Now, what can we say about the integer 1 ? Is it a prime or a composite? Since 1 has the only possible factorization 1’1 we could agree that it is a prime. We might also consider the product 1∙p as a product of two primes; somewhat awkward for a prime number p.—The dilemma is solved if we adopt the convention of classifying the number 1 as neither prime nor composite. We shall call the number 1 a unit. The positive integers may thus be divided into:

  1. 1.

    The unit 1.

  2. 2.

    The prime numbers 2, 3,5,7,11,13,17,19, 23,…

  3. 3.

    The composite numbers 4,6, 8, 9,10,12,14,15,16,…

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Authors and Affiliations

  1. Department of Mathematics, The Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Hans Riesel

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  1. Hans Riesel
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Correspondence to Hans Riesel .

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Riesel, H. (2011). The Number of Primes Below a Given Limit. In: Prime Numbers and Computer Methods for Factorization. Modern Birkhäuser Classics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8298-9_1

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