# The Number of Primes Below a Given Limit

Chapter

## 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.
The unit 1.

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

- 3.
The composite numbers 4, 6, 8, 9, 10, 12, 14, 15, 16,...

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### Bibliography

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