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Nuprl Section: PrimesSquareRoots - Stand-alone sample: Primes have no rational square roots.
Here we give formal proofs along with informal glosses of proofs to the effect that 2 has no rational square root, and more generally, that no prime number has a rational square root.
The proof about 2 is simpler, of course, and so is a better approach to the one about primes.
The method is to show that one could always convert a (non-negative) ratio whose square is two (or prime) into another one with a smaller numerator.