# The Number of Primes Below a Given Limit

• Hans Riesel
Part of the Progress in Mathematics book series (PM, volume 126)

## 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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