gleaming pumice
Unfortunately this won’t make sense if you haven’t seen the “playing billiards with pi” proof or this video by 3b1b: https://youtu.be/jsYwFizhncE
I’m also very tired so forgive any ambiguity
Recently I’ve stumbled upon a conjecture that stated “there exists a number N such that ⌊ π x 10^N⌋is a perfect square” Or, “A linear sequence of N amount of pi’s digits forms a perfect square”. Now, I decided to take a crack at this, using the logic behind the above proof.
First let’s assume N is real. Then we can rewrite the above to be ⌊ π x 100^(N/2) ⌋. Then, we can get a mass ratio of 100^((N/2)-1):1. Using the slope formula, we get a slope of 10^((N/2)-1).
Now, we can substitute this into the formula given, N * (2 θ), where θ is equal to the arc tan of the slope, and N is equal to ⌊ π x 10^N⌋.
Given that N(2 θ) < π < N+1(2 θ), we can do the following:
⌊ π x 10^N⌋(2arctan(10^(N/2-1) < π <⌊ π x 10^N⌋(2arctan(10^(N/2-1) + (2arctan(10^N/2-1)
Does anyone know what to do next? I tried plugging in infinity for N and I just got
10^N * π < 1 < 10^N * π + π which I tried to simplify to 0 < 1 - 10%N * π < π, and then I tried to solve for N but I got -0.1664…. What do I do? It’s 5 am and I’m going insane over this. Thanks !
thick gate