If you take a massive number, divide it by every prime number smaller than itself, and add up all the remainders, what do you get? Because prime numbers are notoriously unpredictable, you might expect the result to be completely random. However, this paper reveals a beautiful hidden pattern. As your starting number gets larger and larger, the sum of these remainders perfectly settles at exactly 35. 5% of the sum of the primes themselves. Even more surprisingly, the exact mathematical constant that dictates this ratio is in terms of π, connecting the chaotic world of prime remainders directly to ππ and the geometry of circles.
Y K Terence Cheng (Mon,) studied this question.