I mean the proofs are wrong, though deceptive enough to have fooled some well established mathematician (to be fair that was before the mathematical concepts were as formal as they are now).

But being unable to provide such positive integers is not a proof that they do not exist. There are statements in mathematics that we know we can't answer. For example, the Continuum hypothesis, which states that every infinite subset of IR is either in bijection with IN or IR. This is known to be unprovable from the standard axioms of set theory, so you won't be able to find any counterexample. But also we can't prove that it doesn't exist.