In the Futurama movie, The Beast with a Billion Backs, one scene features a blackboard with two different proofs of the Goldbach conjecture. The Goldback conjecture is one of the oldest unsolved problems in mathematics. One of those problems, like Fermat’s last theorem or the 4-color problem of maps that is deceivingly easy to state and fiendishly hard to prove: that every even number greater than 4 can be expressed as the sum of two primes.
Just how intriguing this problem is, it’s well illustrated by the following plot that shows the number of ways an even number between 4 and 1 million can be split into a sum of two primes:
Golbach partitioning to 1000000
This plot, taken from Wikipedia, clearly shows that the results cluster along curves (asymptotes? attractors?) that follow some kind of a power law with an exponent between ~0.68 and ~0.77, and there may also be some fractal splitting involved, too. This plot is known as Goldbach’s comet. All I have to do is look at it to understand why many people find number theory endlessly fascinating.