Why is that a conjecture is written and its proof can not be done for years?

Question

Why is it that if a conjecture is written and its proof can not be done for years, means if its statement is understandable in language, why is the proof not possible if the conjecture is correct?
E.g., the proof of Fermat's last theorem (which was very clear what is it saying) was done 350 years later than it was conjectured and it took 130 pages.
Fermat couldn't think for 130 pages and then give the conjecture. So why is the proof so horrible for such simple equation?

Answer 1

There is in fact a precise sense in which this situation can't be avoided: we can prove that (unless ZFC is inconsistent) there is no computable function $F$ such that every ZFC-theorem of length $n$ has a ZFC-proof of length $\le F(n)$.

• Here "ZFC" is just the generally-accepted set of axioms for mathematics (we need to fix some set of axioms to rigorously talk about proofs in the first place). Incidentally, the fact above is itself provable in ZFC (and indeed galactically less).

This is a consequence of Godel's incompleteness theorem: if we had a computable proof-length-bounding function $F$, we could use it to build a computable complete consistent extension of ZFC (this is a good exercise).

So there will always be "short" theorems with only "long" proofs.

(Technically this only addresses the length issue as opposed to the time issue. But it should be clear that the theme is the same: computability-theoretic barriers prevent any sense in which "simple" questions can always have "simple" resolutions, even if we restrict attention to questions which are answerable in the first place.)

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That said, this doesn't address the fact that there are interesting examples of theorems (like FLT) which we naturally think of as much shorter than their proofs. The argument above doesn't rule out the possibility that "natural" mathematical questions always have "not-too-long" proofs. But frankly that's too subjective an issue to be rigorously addressed.