As n gets bigger, the probability of n being a perfect square gets smaller and smaller. In the limit, the probability is zero.
Does this mean square numbers don't exist?
Roughly, it goes as such:
1) the probability of N being a perfect square is proportional to 1/sqrt(N).
2) For any N_0 arbitrarily high, if you integrate from N_0 to infinity the expression (1/sqrt(N) dN), you get infinity.
3) The expression in 2) is the "Feynman equivalent" of the expected number of square numbers above N_0.
So Feynman's nonproof actually turns out to be true, despite it not being a proof in this case as well.
If you integrate the characteristic function of the rational numbers over any interval, you get zero because rational numbers are very rare.
So they don't exist either?
To be less glib, I don't see Feynmann's argument to be bringing anything new. We already knew that counterexamples, if they existed, would be very rare because we tried looking for them with computers and we couldn't find them. But stuff being rare still doesn't prove anything.
Many of us were fooled by Skewe's number:
https://en.wikipedia.org/wiki/Skewes%27s_number
There's no way to conclude that this exists via brute calculation. It's just inconceivably large and would have eluded any of Feynmann's methods.
So I don’t think you have produced a compelling counterexample yet (though I expect you are right that one exists).
It doesn't lead you directly proof - but often just knowing what the answer 'should be' can be a real guiding light.
This kind of thing is used a lot in number theory to figure out the plausibility of some theorem. A lot of open number theory problems are of the form "Prove [unlikely event] never happens."
Edit: Removed braindead argument.
Schwartz distributions, infintesimals; okay, fine, those turned out to be a weird trick that can be formalised. But sometimes their tricks are just plain wrong and this is one example of a trick that just is wrong and can't be formalised.
Consider, many useful primality tests are statistical in nature. It’s pure math, and exact answer is possible but it’s still useful to get a quick check to see if something is a waste of time.
Really, if a full solution takes 20 years you don’t want to actually spend 20 years without having a very good idea it’s going to work.
So I find his small probability, however tiny, that out of all possible integer tuples none of them are a counterexample to be utterly unconvincing and ultimately misguided.
To be clear: my complaint is that there is no way to turn this kind of argument into an actual proof. We could salvage other physicist arguments, but not this one. Probability zero on an infinite set cannot mean inexistence. And the rest of what he's doing, trying to determine that counterexamples must be rare, is "well duh, we knew that, because we've been looking for them."