Good examples for mathemathical problems/statements that are easely solvable/provable in one theory and hard to solve/prove in another

Solution 1:

The prime number theorem states that the number of primes less than a real number $x$ (denoted by $\pi(x)$ ) can be approximated by $x/\log x$ in the sense that $$\lim_{x\to\infty} \frac{\pi(x) \log x}{x} = 1.$$

Now, the very statement of this theorem uses the concept of primality and a limit of a real function, so any proof must, at the very least, use some elementary number theory and some basic real analysis. It turns out that it is indeed possible to prove the prime number theorem with only these basic tools (Selberg/Erdős discovered such a proof) but this proof is (relatively) quite difficult.

It was discovered several decades after the first proofs were found by Hadamard and de la Vallée-Poussin (independently) in 1896. Both of their proofs used complex analysis (in fact, they developed quite a lot of the theory of complex analysis largely with this purpose in mind I recall). So the Prime number theorem is an example of a theorem that can be stated with only elementary number theory and real analysis, and can be proved with much effort with only these theories, but is much more easily established by complex analytic methods.

Solution 2:

Matrix multiplication is associative: you could prove this by a computation, but it's obvious once you realize a matrix gives a function $\mathbb{R}^n \to \mathbb{R}^n$. This one is sort of cheating, because it's not clear why you'd want to multiply matrices unless you understood this property already.

Subgroups of free groups are free: this can be shown group-theoretically but is not obvious. Topologically, this says covers of graphs are graphs, which is obvious.

Hilbert's Theorem 90 can be proved directly in a couple paragraphs, using linear independence of characters, but spotting the theory of etale cohomology it just says that there are no non-trivial line bundles on a point.

Solution 3:

1. Fundamental theorem of Algebra:

   Algebraic: I know of two proofs, one using induction and doing a whole lot of algebraic manipulations that made it quite lengthy. Another one uses Galois theory.

   Complex Analytic: In my opinion some real cool proofs arise from complex analysis. The most elegant( IMO) is the proof using Liouville's Theorem though proofs using minimal modulus principle is also quite good.

2. Finding real integrals using contour integration.(Has been mentioned before by amWhy)

Right now these two stike my mind.