(with programming involved)
Many thanks to Professor Rafe Jones. You can see this in all its glory here (on Rafe Jones' page) and read the number theory paper this is based of off here (Rafe's Paper)

#Number Facts

Sophie Germain Prime:

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 29 is a Sophie Germain prime and 2 × 29 + 1 = 59 is its associated safe prime.

http://en.wikipedia.org/wiki/Sophie_Germain_prime

Happy Number:

A happy number is a number defined by the following process: Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.

http://en.wikipedia.org/wiki/Happy_number

Prime Number:

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.

More to come

http://en.wikipedia.org/wiki/Prime_number