BILL: Justin, you've seen a proof of VDW's theorem, but there are easier things you haven't seen and should. Can you prove that the number of primes is infinite?
JUSTIN: Thats easy.
BILL: Good. How does it go?
JUSTIN: By the Green-Tao Theorem there are arbitrarily long arithmetic sequences of primes. Hence there are an infinite number of primes.
What to make of this?
- Justin does not know how to proof the Green-Tao theorem (neither do I). However, if the proof does not use the fact that there are an infininte number of primes, then Justin's proof is valid. READERS: does anyone know, does it use the infinitude of the primes?
- Justin now knows the standard proof. However, he should also learn that, at the level of math where he is at, you should be able to prove everything you use.
- When does one start using theorems whose proofs one does not know? In research this is common. For basic coursework it should be rare.