# Prime Numbers Essay

Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians.

The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers.

A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28.

A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other and vice versa.

You can see more about these numbers in the History

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He proved what has come to be known as Fermat's Little Theorem (to distinguish it from his so-called Last Theorem).

This states that if p is a prime then for any integer a we have ap = a modulo p.

This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2n - 2 is divisible by n. The other half of this is false, since, for example, 2341 - 2 is divisible by 341 even though 341 = 31 11 is composite. Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers.

Fermat corresponded with other mathematicians of his day and in particular with the monk Marin Mersenne. In one of his letters to Mersenne he conjectured that the numbers 2n + 1 were always prime if n is a power of 2. He had verified this for n = 1, 2, 4, 8 and 16 and he knew that if n were not a power of 2, the result failed. Numbers of this form are called Fermat numbers and it was not until more than 100 years later that Euler showed that the