Theorems with simple proofs with one method of proof, and incredibly difficult with another. This is a very soft and potentially naive question, but I've always wondered about this seemingly common phenomenon where a theorem has some method of proof which makes the statement easy to prove, but where other methods of proof are incredibly difficult.
For example, proving that every vector space has a basis (this may be a bad example).  This is almost always done via an existence proof with Zorn's lemma applied to the poset of linearly independent subsets ordered on set inclusion. However, if one were to suppose there exists a vector space $V$ with no basis, it seems (to me) that coming up with a contradiction given so few assumptions would be incredibly challenging.
With that said, I had a few questions:

*

*Are there any other examples of theorems like this?

*Is this phenomenon simply due to the logical structure of the statements themselves, or is it something deeper?  Is this something one can quantize in some way? That is, is there any formal way to study the structure of a statement, and determine which method of proof is ideal, and which is not ideal?

*With (1) in mind, are there ever any efforts to come up with proofs of the same theorem using multiple methods for the sake of interest?

 A: Questions like this always remind me of Hamiltons search for a multiplication in $\mathbb{R}^3$ which somehow extends the multiplication of real and complex numbers (*). He searched, in vain, for years. And he was, at his time, a really famous and renowned mathematician.
Today it is easy to see that this is not possible. If a vector space $V$ with odd dimension is a division algebra, then for  $0\neq a\in V$ the map $x\mapsto ax$ is a linear map which must have a real eigenvalue $\lambda$ (due to the mean value theorem). If $v\neq 0$ is an eigenvector we have $(a -\lambda) v =0$, so $a= \lambda e$. Since $a$ was chosen arbitrarily $V$ must be isomorphic to $\mathbb{R}$.
What Hamilton was lacking were the concepts and definitions involved in this short proof. There is a lot of power and hidden knowledge and the result of decades of research effort in the definitions we are taught today when we learn mathematics.
((*) Summarized from the introduction to an article by Köcher and Remmert, from the book 'Numbers' by Ebbinghaus, Hermes, Hirzebruch et al., German version, Springer 1983)
A: *

*Yes, there are many other examples of this. More or less every famous result (well, you wanted a "simple proof" for it) will be "incredibly challenging", or even impossible with a different proof. I think of Fermat's last theorem, the Poincare conjecture, or the weak Goldbach conjecture, just to name a few. Of course, the word "simple proof" depends on the context. Perhaps one day the proof for FLT is considered to be "simple" in comparison to other proofs.


*No, I don't think this is apparent from the statement alone. Take Fermat's last theorem. How could one "quantise" this beforehand, that a solution without elliptic curves and without modular forms will be (probably) extremely challenging, and much more difficult than the proof we have?


*Yes, they're  famous theorems, where people have tried to find as many proofs as possible. Three examples are the Pythagorean theorem, the quadratic reciprocity law, or the fundamental theorem of algebra.
The wikipedia article here mentions that " Several hundred proofs of the law of quadratic reciprocity have been published."
A: Very often the first proof of a result which appears in the literature is extremely messy because the mathematician who proved it is working at the very edge of what is possible with the tools of the day; then it gets simplified over time as other mathematicians better understand what is going on and develop better machinery for streamlining the proofs. These first proofs are typically not presented to students because they are terrible, but the disadvantage of not knowing them is that you don't see how valuable the machinery that streamlines the modern proofs is.
There are many examples of this sort of thing, some of which you can find at this MO question; here's one that I came across while writing a blog post about the Sylow theorems. It is about

Cauchy's theorem: if a finite group $G$ has the property that its order $|G|$ is divisible by a prime $p$, then $G$ has an element of order $p$.

There is an extremely slick proof of this theorem which comes from consider the set of solutions to the equation
$$\{ (g_1, \dots g_p) \in G^p : g_1 g_2 \dots g_p = e \}$$
and then considering the action of the cyclic group by rotation $(g_1, g_2 \dots g_{p-1}, g_p) \mapsto (g_2, g_3, \dots g_p, g_1)$, which you can see in the link. It takes maybe three sentences to give.
By contrast, Cauchy's original proof took 9 pages. He does it by explicitly constructing the Sylow $p$-subgroups of the symmetric group, then (I believe Cauchy was working at a time when "finite group" always meant "finite group of permutations" so for him all finite groups were already embedded into symmetric groups) using a clever counting argument to show that if a finite group $G$ has the property that $p \mid |G|$ and also embeds into another finite group which has Sylow $p$-subgroups, then $G$ has an element of order $p$; you can see the details in the link. I give a very abbreviated sketch of the proof; the full construction of the Sylow $p$-subgroups of the symmetric group is very tedious (I have never seen anyone give it in full, and tried doing it in a follow-up blog post but gave up because it was too tedious).
This is a good example of what I mean; Cauchy was working at a very early time in group theory before anyone had even defined an abstract group, and people just didn't understand group theory that well yet. There was not even the notion of a quotient group at the time. Once group theory was better understood better proofs were possible. Actually I have no idea who the above slick proof of Cauchy's theorem is due to nor how many decades it took after Cauchy's original proof for someone to find it.
Cauchy's original proof does have the advantage that it is much closer to being a proof of the first Sylow theorem. It has a generalization due to Frobenius which shows that if a finite group $G$ embeds into a finite group $H$ which has a Sylow $p$-subgroup, then $G$ must have a Sylow $p$-subgroup. And then you can prove Sylow I by exhibiting the Sylow $p$-subgroups of the symmetric groups, or somewhat more easily, the general linear groups $GL_n(\mathbb{F}_p)$, then invoking Cayley's theorem.
A: Burnside's theorem that every finite group of order $p^aq^b$ (where $p,q$ are primes) is solvable has a short proof using character theory and a much longer proof without characters. See M. Isaacs' books for both proofs.
