Some numerology

Imagine that you numerically calculate

• exp(pi*sqrt(67)).

You get some large number. Remove the integer part of it. What's left? Let me tell you that it starts with a couple of digits "9". How many? Well, the answer turns out to be six. Pretty close to an integer.

You say. This had to be an accident. Let's try another example:

• exp(pi*sqrt(163)).

Subtract the integer part. What would you expect? Well, the answer turns out to be

• 0.99999999999925

starting with twelve digits 9. Those who thought that it was an accident in the first example have suffered a heart attack after this second, much more spectacular example, and died away. Surely, it can't be a coincidence. Only six characters are needed to define the number and it gives you twelve times 9. Indeed, it's not a coincidence. It's because if you substittute an appropriate argument to the j-function, you may derive that the number is close to an integer with some polynomial tricks.