Elementary problems that would've been hard for past mathematicians, but are easy to solve today? [closed]

Solution 1:

I would say that computing the Fourier coefficients of a tamed function is a triviality today even at an engineering math 101 level.

Ph. Davis and R. Hersh tell the long and painful story of Fourier series. I quote from their book:

"Fourier didn't know Euler had already done this, so he did it over. And Fourier, like Bernoulli and Euler before him, overlooked the beautifully direct method of orthogonality [...]. Instead, he went through an incredible computation, that could serve as a classic example of physical insight leading to the right answer in spite of flagrantly wrong reasoning."

(Fifth Ch. "Fourier Analysis".)

Solution 2:

That there exist transcendental numbers. This was first shown by Liouville, who proved that Liouville's number: $$\sum_{i=0}^\infty10^{-i!}$$ is transcendental.

The "modern" proof would be due to Cantor:

There are countably many algebraic numbers and uncountably many reals. Therefore there exists a transcendental number.

Proving that Liouville's number is transcendental isn't so hard, but compared to the above it seems quite torturous.

Solution 3:

This sum-of-squares theorem of Fermat may qualify as an example:

An odd prime $p$ is expressible as the sum of squares $x^2+y^2$ if and only if $p\equiv 1 \text{ mod } 4$.

You can read this Wikipedia article (as of the most recent update to this answer) to see the difference in mental effort in the original proof by Euler, as opposed to a modern treatment using the fact that the Gaussian integers are a Euclidean domain.

A dual example: I think Brouwer would be astonished and pleased to know that the Brouwer fixed point theorem can now be proven for the simplex (and, with more effort, for convex polytopes) with absolutely no knowledge of topology; just some affine geometry and combinatorial intuition to prove Sperner's Lemma, and basic analysis to translate to the continuous setting.

It's still not an "easy" proof but it is an example of a classical problem that we now can solve with considerably less machinery, instead of the above example, whose ease of proof can be chalked up to more machinery.