Past open problems with sudden and easy-to-understand solutions

Solution 1:

The integral of $\sec x$ stumped mathematicians in the mid-seventeenth century for quite a while until, in a flash of insight, Isaac Barrow showed that the following can be done:

$$\int \sec x \,\mathrm{d}x= \int \frac{1}{\cos x} \, \mathrm{d}x=\int \frac{ \cos x}{\cos^2 x} \, \mathrm{d}x=\int \frac{ \cos x}{1-\sin^2 x} \, \mathrm{d}x.$$

Using $u$-substitution and letting $u=\sin x$, the integral transforms to

$$\int \frac{1}{1-u^2} \, \mathrm{d}u,$$

which is easily evaluated by partial fractions.

Solution 2:

Theorem: transcendental numbers exist and there are (uncountably) infinitely many of them.

The existence of transcendental numbers had been conjectured for over 100 years before Liouville constructed one in 1844. Other numbers such as $e$ were shown to be transcendental one by one. Cantor was able to prove their existence with ease:

Proof: the algebraic numbers are countable and the real numbers are uncountable.

Solution 3:

The $\mathcal{AKS}$ (Agrawal, Kayal, Saxena) algorithm, which proves that we can answer if a number is prime or not in polynomial time. It has been found in 2003 and is said "reachable by ordinary man" in reason of the background it needs to be understood. More info here (wiki) and here (the paper).

Solution 4:

It is not completely elementary, but Abel's proof of the Abel-Ruffini theorem is quite short, 6 pages, and can with a bit of introduction be understood by someone without a degree in mathematics. The Abel-Ruffini theorem states that there is no general solution in radicals to a degree 5 or higher polynomial equation.

The Abel-Ruffini theorem had been open for over two hundred years and was one of the central problems in mathematics of that time, akin to the Riemann Hypothesis now. For degree 2, a formula had been known since 2000 BC to the Babylonians. For degree 3 and 4, formulas had been discovered 200 years earlier. The search for a formula of degree 5 had been long in progress.

Solution 5:

Quadratic reciprocity.

Euler has stated the theorem but never managed to prove it, and it took Gauss many years to prove this theorem, and right now we have over 200 different proofs, some of which could be explained in an hour long lecture.