# Theorems which later turned out to be vacuous

Has it ever happened that a theorem of the form

If $P$, then $Q$

was proven and published, perhaps with great difficulty, only for someone to realize later that the condition $P$ of the theorem is never satisfied, or, worse, that the conclusion $Q$ of the implication is false?

For example, if the Generalized Riemann Hypothesis were disproved tomorrow, I would have a large supply of examples on my hands, as so many results are conditional on GRH. But surely, this must have happened before, in the long history of mathematics?

• You make it sound as if the proof of "if P then Q" was wasted effort. But maybe the proof of the stronger result "if P then 0 = 1" is based on the proof of the partial result "if P then Q"? – bof Nov 18 '14 at 22:47
• Yes, I agree with bof. If $Q$ is known to be false, then proving $P\rightarrow Q$ gives you some useful information that is exactly what you mention: $P$ never being satisfied. – Hayden Nov 18 '14 at 22:50
• I agree with both of you that many interesting things could still arise out of such a situation, all of which could add to the charm of such an example. Presumaby, the effort would be completely wasted only if $Q$ were later proven to be false. – Bruno Joyal Nov 18 '14 at 22:54
• Possible urban myth, S. T. Yau once said (he really said it, that's not the myth part, I was there) that there had been a Ph. D. dissertation on Holder continuous functions with exponent above $1.$ The trouble being that all such functions are constant. – Will Jagy Nov 18 '14 at 22:57
• The second example in this answer seems to qualify. – Semiclassical Nov 19 '14 at 0:03

In the proof of FLT, was essential the fact that nontrivial FLT solution $\implies$ $\exists$ weird elliptical curve. Wiles proved that there isn't weird elliptical curve. See A question on FLT and Taniyama Shimura.