The most remarkable thing is that this is actually not a coincidence, but a consequence of complex multiplication: https://en.wikipedia.org/wiki/Complex_multiplication#Sample_...
extremely close to an integer
it equals 262537412640768743.99999999999925...
The chance that a random number is that close to an integer is about 1:10^12.
So by trying 10^12 crazy formulas you will have one hit. 10^12 possibilites can be covered in about 40 bits.
If we look at "e^(pi*sqrt(163))" as 16 ascii characters, that's 112 bits.
Every integer has a unique prime factorisation: that's something kids learn in primary school. The idea of factorisation goes a lot deeper though. Define the gaussian integers as integers of the form a + ib, where a and b are intgers. Unique factorisation holds in the Gaussian integers too, but some of the numbers we recognise as prime are no longer prime. For example, 2 = (1 + i)(1 – i).
It turns out unique factorisation is a rare property of number systems. Consider the set of numbers whose ‘integers’ are of the form a + b sqrt(-5), with a and b standard integers. In this system, 6 = 2x3 = (1 + sqrt(-5)) x (1 - sqrt(-5)). But none of those factors can be factorised further, so 6 has two distinct factorisations.
It turns out there are only nine positive numbers n so that unique factorisation holds in the system a + b*sqrt(-n). 163 is the largest!