A Prime Number is basically a number that has only two factors: 1 and itself. (A factor, by the way, is a whole number that divides evenly into another number.)
In his book, Mathematical Mysteries, Calvin C. Clawson tells us that every number in the natural number system can be defined as either prime or composite (made up of several primes). The number 1, however, is not considered a prime number, even thought it satisfies the requirements for "primeness". This is because, like a nuclear bomb, it would destroy the "Fundamental Theory of Arithmetic", which stipulates that any number can be "decomposed" into a set of unique prime factors. (Such a catastrophe would have universe shattering repercussions in the field of mathematics).
The primes within the first hundred 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.
We can tell whether a number is prime or not first by inspection. First of all, no even number except 2 may be a prime because 2 divides evenly into every even number. Then we can divide the number we are inspecting for primeness by every prime number beneath it up to half of its amount (half because no number higher than the half way point could possibly divide evenly into our number, and prime numbers because as noted above all composite numbers are composed of prime factors).
But this might seem a cumbersome way to find prime numbers. Edward Waring in the 1700s actually discovered a way to test a number for primeness. He called it Wilson's Theorem after a friend. His theorem runs thus, "A number is prime, if and only if,(P-1)!+1 is divisible by the number in question (P). The symbol ! means "factorial", which is another way of saying multiply together all the numbers up to and including the number preceding the symbol(for example: 5!=1X2X3X4X5). The problem is that even this theorem and the resulting equation is quite cumbersome. Imagine having to multiply all the numbers together up to and including say a million! How about numbers with thirty or forty digits!
Euclid, the famous geometer, proved that there are an infinite number of prime numbers. He did this by assuming that if there are a limited number of primes that there must be a greatest prime number. He showed that this was impossible by constructing a number. This number Clawson, in his book, Mathematical Mysteries, designates P. Euclid then constructed a number that multiplied all the primes together up to P and then added 1. The problem here is that if you take this number and divide it by any combination of primes it will always leave a remainder of 1, meaning this number itself must be prime. Since there can be no largest prime, there must be an infinite number of them within the universe of natural numbers.
As the infinite number of primes get larger as individual numbers, they tend to be spaced farther and farther apart, so that they become relatively rare. It is thought to be impossible to create a formula to calculate primes. Mathematicians are constantly at work seeking larger and larger primes. They are useful in "real world" applications primarily in cryptology (the study of encoding).