[FOM] Prove something weaker!

[joeshipman at aol.com]

I'm interested in the phenomenon of famous conjectures being stated in strong forms when even very much weaker forms of them are unproven.


For example, I don't understand why I see a lot more talk about the inability to prove that P is not NP, when the inability to prove that P is not PSPACE seems far more surprising.


Similarly, the abc conjecture says that for any x>1, there are only finitely many integer triples a+b=c satisfying c>rad(abc)^x where rad(N) is the product of the distinct primes dividing N, but I'm not aware that this has been proved for ANY x, let alone all x>1. In fact, I'm not even aware that a much, much weaker statement than the existence of some such x has been proven, namely that for any finite set S of primes, there are only finitely many integers of the form n(n+k) all of whose prime factors are  in S. Or even weaker, take k=1!


I know that sometimes it is easier to prove something by proving a generalization of it, but that can't be a sufficient explanation for how much I see this. Is it that mathematicians wish to be seen as trying to prove something difficult and non-obvious, rather than trying to prove something that everyone is certain is true but no one can prove? From my point of view the latter type of discovery is much more valuable, because the existence of a deep gulf between our intuition and our knowledge is much more troubling than the existence of difficult propositions whose truth value we don't have general agreement on.


Can anyone supply a technical reason that separating P and PSPACE shouldn't be easier to tackle than separating P from NP? Or a technical reason why progress on the much weaker number-theoretic questions above is unlikely without settling the abc conjecture first?


I'm also interested in collecting other examples of well-known conjectures for which there are much weaker versions that seem almost equally difficult to make progress on, besides the two examples above.


-- JS
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