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:
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.
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