Mathematicians have a special fondness for definitions. The objects they study often have no physical presence: you can't touch them, weigh them, or smell them. So, precise definitions are used to provide a solid underpinning to their subject. Definitions are made for two reasons. Sometimes a mathematician makes a definition in order to have a simple shorthand term for a more complicated object that was needed to make sense of a proof. In other cases, a definition is a kind of generalization, made to unite lots of examples and special cases into a single kind of thing deserving further study. In the present document, we want to investigate some of the consequences of a single, simple definition of the latter kind.

**Definition:** An integer `p` is called a
*prime* number if the only positive integers that divide `p` are `1` and
`p` itself. Integers that are not prime are called
*composite*.

Here are the first few prime numbers:

$2,\; 3,\; 5,\; 7,\; 11,\; 13,\; 17,\; 19,\; 23,\; 29,\; 31,\; ...$

Prime numbers are special because they are the elementary building blocks of the multiplicative structure on the integers; every integer can be written in only one way as a product of its prime factors. The mathematically precise version of this assertion is known as the The Fundamental Theorem of Arithmetic.

There are a number of interesting questions you can ask about prime numbers and factorizations.

- Given any integer
`n`, how do you find its prime factorization? - Given any integer
`n`, how do you decide if it is a prime number? - How many prime numbers are there?
- Can you list all the prime numbers less than a given bound?

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Kevin R. Coombes

Department of Biostatistics

University of Texas M.D.Anderson Cancer Center

1515 Holcombe Blvd., Box 447

Houston, TX 77030