TITLE: Use of Conjectures to Prove a Theorem
QUESTION [11 upvotes]: Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that is independent of C. For instance, it would be a claim that can be proved if P=NP, and can also be proved if P is different from NP.
I apologize if the question was previously asked (I also apologize for the title if it does not faithfully reflect the question).

REPLY [31 votes]: Another standard example is Littlewood's theorem that the number of primes less than x is sometimes greater than Li(x). His proof used different arguments depending on whether the Riemann hypothesis is true or false. See http://en.wikipedia.org/wiki/Skewes%27_number

REPLY [22 votes]: How about this basic one:
Theorem: There exists an irrational number $p$ and an irrational number $q$ such that $p^q$ is rational.
Proof: Let $q=\sqrt{2}$ and note that it is irrational.  Conjecture: $q^q$ is irrational.  If this conjecture is false, then we are done (the theorem is proved with $p=q$).  If the conjecture is true, then let $p=q^q$ and note that it is irrational.  The theorem is true in this case as well since $$p^q=(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=\sqrt{2}^2=2$$ is rational.