Here are few of the more common and simple theories that relate to prime numbers.

Goldbach's Conjecture

Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states that: Every integer greater than 2 can be expressed as the sum of two primes.

This stems from 7th June 1742, when German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) in which he proposed that every integer which can be written as the sum of two primes.

Mersenne Primes

Marin Mersenne, a French monk who began the study of Prime numbers in the early 17th century, noted that some primes can be found by:

A positive integer that is one less than a power of two.

In notation this would be:

M=2p-1 where P is prime.

For example:

M=25-1 where P is prime.

M=(2 x 2 x 2 x 2 x 2) - 1

M = 32-1

M=31

Note that while this formula can be used to predict some Prime numbers is does not always hold that the number found is actually prime.

To date only 49 Mersenne primes have been found, so they are pretty scarce. Since 1997, all newly-found Mersenne primes have been discovered by the "Great Internet Mersenne Prime Search" (GIMPS), a distributed computing project. The 49th Mersenne prime was found on 7th Jan 2016 by Dr. Curtis Cooper and is 274,207,281-1, a 22,338,618 digit number.

Riemann Hypothesis

The Riemann Hypothesis, is one of the Clay Mathematics Institute's "Millenium Problems" and is currently unsolved. The Riemann Hypothesis concerns the distribution of prime numbers. Even with today's sophisticated maths we have not found a formula that allows us to define the distribution of prime numbers.

In 1859 Riemann showed that the distribution of prime numbers was related to a function called the Riemann Zeta function. To solve the Clay Mathematics Institute's Millenium Problem related to the Riemann Hypothesis you simply need to demonstrate that "the real part of any non-trivial zero of the Riemann zeta function is 1/2". To win the prize you can either prove or disprove the statement.

To date the first 100 billion numbers or so have been shown to be on the same line but nobody has proved that this holds for all "non-trivial" zeros.

If you want to know more there is a Numberphile Youtube video that does a pretty good job of explaining the problem for lay mathematicians.