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

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Let $P$ be a mathematical statement or a mathematical problem. I am looking for a couple of nice examples for $P$ that satisfy the following criteria:

  1. Given two (or more) mathematical points of view on $P$, we find that one of them views makes $P$ easy to solve/prove and the other one makes $P$ hard to solve/prove.

  2. $P$ should (at least from the easier point of view) be understandable by someone who studied maths obtained the basics and has a quiet good understanding of mathematical problems.

  3. It shouldn't take to much text to formulate, since I don't have that much time and space to present it.

It would be nice if the problem is prominent and it is ok if the problem is not pure math but must have a clear link to maths.

Any ideas? A short explanation of the problem from the different angels is welcome and appreciated.

Edit: I forgot to mention that by different point of view I meant somethink like looking at $P$ from an algebraic point of view and from an analytical point of view and maybe from a topological point of view. I want to point out the awesome properties of maths to transform a hard problem to another theory where the problem is easily solvable.

Best Answer

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.

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