Patterns in Prime numbers, and the null hypothesis I've read about many attempts to find patterns in prime numbers.  First, is there a mathematical way to prove there is not a pattern to prime numbers?  Since there are ways to check if a number is prime or not, can these methods be combined into some large function that produces prime numbers?  And in case it isn't obvious, no, I never got passed pre-Cal.
 A: 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
A: 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.
A: 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
