At the risk of sounding like a broken record: Dave Richeson has had some more great posts recently. These two in particular left me in awe of mathematics. Both results I had seen before, but not proved in these ways. These two proofs are so unique. Unfortunately, for my students, both of these particular proofs also require math typically not covered in high school. But that shouldn’t stop you from checking them out. In fact, both results have much simpler and more accessible proofs that I’ve linked to.
1. Irrational rotations of the circle and Benford’s law. Benford’s law is amazing. It predicts, for instance, that in a large list of spread out data, approximately 30.1% of the data will start with a “1”. The other digits have predictable frequency as well. This law allows analysts to detect fraud in data. Cool! Richeson proves Benford’s law for the powers of 2 (so…not a full proof of Benford’s) using irrational rotations of a circle. For more on Benford’s and a more straightforward justification, read the wikipedia article or one of the other sources Richeson links to.
2. Hillel Furstenberg’s proof of the infinitude of primes. There are infinite primes. For other proofs, see this page, especially the classic number-theoretic proof by Euclid. Euclid’s proof is definitely accessible to high school mathematicians, and it’s pretty elegant and exciting if you haven’t seen it before.