The Cyclic Constant Theorem

The Universal Theorem Take any prime number (except 2 or 5 if you're in base 10, but really any prime that doesn't divide your base). Look at the repeating decimal expansions (in whatever base you choose) for all the fractions from 1 over that prime up to (p-1) over that prime. Each of these fractions repeats with some period — call it r. Now pick any way of chopping that repeating block into equal-sized chunks, where the chunk size divides evenly into r. For each fraction, add up those chunks (treating them as whole numbers), then divide by a string of repeated digits in your base — like 9, 99, 999 in base 10, or 1, 11, 111 in base 2, etc. This gives you a number — call it the "multiplier" for that fraction. --- Here's what the theorem says: If you add up all those multipliers — for every fraction from 1/p up to (p-1)/p — the total is always: (p − 1) × r ÷ (2k) where: · p is your prime · r is the length of the repeating cycle (the period) · k is the size of the chunks you chopped the digits into --- What Makes This Beautiful · The long string of repeated digits you divided by? It completely disappears. The final answer doesn't depend on it at all. · It works for any base — base 10, base 2, base 16, whatever. · It works for any prime that doesn't divide the base. · The digits can dance however they want — the total across all fractions is fixed.