Patterns in Prime numbers, and the null hypothesis

I like Dirichlet's Theorem, which states that for relatively prime $a,d \in \mathbb{Z}^+$, there are infinitely many primes in the progression $\{a + nd \mid n \in \mathbb{Z^+}\}$. Further, the proportion of primes in any relatively prime residue class of $d$ is about $1/\phi(d)$, where $\phi$ is the Euler phi function.

In essence, the primes can be seen as somewhat evenly distributed. You can see more below.

http://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions

I especially like this characterization of primes made by Don Zagier

"There are two facts about the distribution of prime numbers which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts.

The first is that despite their simple definition and role as the building blocks of the natural numbers, the prime numbers... grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout.

The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, and that they obey these laws with almost military precision." Don Zagier, Bonn University inaugural lecture

This is taken from this site where you can find many more quotations from prominent mathematicians.

Here you can find some formulas for primes.

It really depends on what you mean by patterns. Legendre showed that there is no rational algebraic function that outputs only primes. You can read more about prime generating functions here

http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

The famous Green-Tao theorem states that there are arbitrarily large arithmetic progressions in primes

http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem