Order out of noise — page 6
Number
Which way a prime's two squares point
A prime one more than a multiple of four is a sum of two squares in exactly one way, and the two squares make a point on a circle. Draw that point for every such prime and ask which way it faces. It faces every way equally — the angles spread evenly over the sector, which Hecke proved by giving each angle a remainder and copying Dirichlet. Measured over seventy-four thousand primes, they are even more evenly spread than random angles would be.
Topology
Every loop of the butterfly links every other
The Lorenz equations have infinitely many periodic orbits, each a closed loop winding round the two lobes in its own order. Name each loop by that order — LR, LLR, LLRLR — and the linking number of any two can be read off the names, without solving anything: it is half the number of times a left-going strand passes a right-going one. Every such crossing has the same sign, so every pair of loops links, positively. Computing forty-five pairs from the equations themselves, the Gauss integral lands on the predicted whole number every time.