F = (2221564096 + 283748 sqrt(462)) / 491993569
plays a fundamental role in number theory!
For any irrational x, we define its 'Lagrange number' to be the supremum of c such that
|(p/q) - x| < 1/cq²
has infinitely many solutions for rationals p/q. So, the bigger the Lagrange number is, the easier x is to approximate by rational numbers. Quite famously, the golden ratio has the smallest possible Lagrange number, namely √5.
Here's the shocking fact: every real number ≥ F is a Lagrange number, and F is the smallest number with this property!
F is called 'Freiman's constant', because he proved this fact. His proof is 100 pages, and I don't want to read it... but some people have.
There's a lot more crazy stuff about the set of all Lagrange numbers. A tiny bit is here:
https://en.wikipedia.org/wiki/Markov_spectrum#Lagrange_spectrum
