Looking over my post again, after a good night's sleep, I see that it wasn't as coherent as it appeared to me yesterday. Let me see if I can put my point a little more clearly.
The paragraph centers its claim of the irrationality of √2 on the idea that p² contains exactly twice as many powers of 2 as p does. But that is only true because √2 is irrational, making the demonstration a circular proof.
Consider. If √2 were rational, in the form of z/y for some coprime integers z and y, then it would be easy to find an integer that is not itself an integer power of 2, but whose square is an integer power of 2; z would be such a number.
But [p^2 contains exactly twice as many powers of 2 as p does] is only true because √2 is irrational, making the demonstration a circular proof.
Notice that you are claiming that all possible proofs of the statement "p^2 contains twice as many powers of 2 as p" require asserting without proof that sqrt(2) is irrational.
Why does the prime factorization of integers depend upon something that is, if not irrational, at least certainly not an integer? (Proof: 1^2 = 1, 2^2 = 4, and x <= x^2 by induction.)
Here's the new thread for posting quotes, with the usual rules: