Amicable Numbers

A pair of integers, each of which is the sum of the distinct proper factors of the other.

Wilson's Theorem

If, and only if, n is a prime number then (n − 1)! + 1 is a multiple of n:

(n−1)! ≡ −1 (mod n)

Where the symbol ≡ means modulus

Euclid

Euclid's Proof that the Set of Prime Numbers is Endless

The proof works by showing that if we assume that there is a biggest prime number, then there is a contradiction.

We can number all the primes in ascending order, so that P1 = 2, P2 = 3, P3 = 5 and so on. If we assume that there are just n primes, then the biggest prime will be labelled Pn . Now we can form the number Q by multiplying together all these primes and adding 1, so

Q = (P1 × P2 × P3 × P4... × Pn) + 1

Now we can see that if we divide Q by any of our n primes there is always a remainder of 1, so Q is not divisible by any of the primes.

But we know that all positive integers are either primes or can be decomposed into a product of primes. This means that either Q must be prime or Q must be divisible by primes that are larger than Pn.

Our assumption that Pn is the biggest prime has led us to a contradiction, so this assumption must be false, so there is no biggest prime.

Goldbach's Conjecture

The conjecture that every even number (greater than or equal to 6) can be written as the sum of two odd prime numbers.

Goldbach also conjectured that all odd numbers are the sum of three odd primes: Vinogradov's theorem shows this true of all except possibly finitely many odd numbers.