Definition of a prime, and some properties.
A prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC.
Prime numbers are good for teaching because they are easy to understand initially, and have many applications at all levels of difficulty. Some exemplar questions:
* How many prime number factors does 10 have? (or any other number)
** Find me another number with the same number of prime factors.
** Find me another number with the same number of factors.
** Which number less thatn 40 (say) has the highest number of factors?
** Can you get me a formula for the number of factors which any number has?
The importance of primes is that any number has a _unique_ decomposition in terms of primes.
The notion of primes can be extended to other sets e.g. to knots, or to the set of posigive even numbers. However, in some of these the prime decomposition is not unique.
Prime numbers are used in coding theory (e.g. for ensuring security on the internet). Thei use there depends upon the fact that it is much easier to multiply to primes together, than it is to do the opposite (i.e.to say which two primes multiply to give a particular number). For example, you can probably multiply 91 and 17. But if I ask you which two numbers give 1547, you may find that much more difficult.
Prime numbers can be represented geometrically. (Think of a composite or non-prime number, which can be represented as a rectangle.) Prime numbers are precisely those which can NOT be represented as a rectangle (except as a trivial rectangle, in which one of the dimensions is of size one).
A further interesting geometric representation of prime numbers is as a _spiral_ (often called the "Ulam Spiral", after Stanislave Ulam). This uses a piece of square graph paper. Put '1' in the middle somewhere. Then put the numbers 2 to 9 around it in order. Now move out one and put in the numbes 10 to 25. (Obvious questions: 9, 25 ... What comes next?) Now carry on spiralling out. When you have reached infinity (or before), start filling in the squares which correspond to prime numbers. What do you notice .....? (The same can be done in the regular multiplication tables using different number bases.)
The first thirty prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113
(sequence A000040 in OEIS)
See the list of prime numbers for a longer list. The number one is by definition not a prime number; see the discussion below under Primality of one.
The property of being a prime is called primality, and the word prime is also used as an adjective. Since two is the only even prime number, the term odd prime refers to all prime numbers greater than two.