# Lessons on Lessening

During your school years, you probably would have seen this equation: $3^2 + 4^2 = 5^2.$

Your teacher most likely would have shown you this whilst rambling on about triangles or something. And maybe, if you were well behaved and pretended to be interested, you might have seen this one: $3^3 + 4^3 + 5^3 = 6^3$

The first equation shows that it is possible to add up two squares to get another square. The second equation demonstrates that you can add up three cubes to get another cube.

Euler, that guy whose name your friends like to pronounce incorrectly, conjectured that at least $n$ $n$th powers are required to sum to an $n$th power. That is, for example, you would not be able to find less than 7 powers of 7 that add up to a power of 7.

Euler was wrong, which is not something you get to say that often. This can be seen in the following ridiculously short paper. Bam. They managed to find a power of 5 that can be written as the sum of only 4 powers of 5, and in doing so they disproved a 200 year-old conjecture and didn’t even hang around to brag about it.

Anyway, the result itself isn’t that interesting (unless you’re a drooling number theorist like me). I mostly enjoyed the fact that the paper was nice and short.

I then went looking for even shorter mathematical papers and found that the community had already burned plenty of tax-funded joules on producing bite-sized journal articles. The following work of Don Zagier is state-of-the-art in this respect. 