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Given a SAT instance. If one knows that there are exponentially many solutions to that SAT instance, then can one find even one solution in polynomial time?

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Could you elaborate on what you mean by "exponentially many solutions"? –  MJD Feb 1 '13 at 22:48
    
possible assignments that satisfy the SAT instance –  J.A Feb 1 '13 at 22:53
    
The problem is that "exponentially" is poorly defined, not the number of solutions, as is the question itself. Finding a solution to a sat problem is a constant time problem or an exponential problem depending on how you define it... it's constant time with respect to the individual SAT instance (though possibly a large constant) and exponential with respect to the size of the sentence. Questions loosely about intractability and computational complexity are related but not automatically identical. –  dezakin May 1 at 23:59

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