Are you familiar with fibonacci? Specifically the number series?
Want to learn something new about it?
If you can calculate the fibonacci sequence I can show you something new.
Did you know that fibonacci repeats inside of itself infinitely?
As the numbers grow, they start appearing all over again? Encoded, yet there to see?
Yes, God. God is everywhere in math. Everywhere in numbers.
It's something that holds true for ALL linear recurrences. The recurrence repeats infinitely inside it self. It's not limited to fibonacci, but that's the obvious one.
Calculate fibonacci(3 * 2^k + 1 + i) (for some k) and compute the LOGICAL AND with fibonacci(i). Do you see it? Fibonacci is encoded inside fibonacci in binary. It repeats infinitely forever.
Yes and no. Old news, yes, but actually "touched"? No. Unknown knowledge, shared first here.
Fibonacci is encoded INSIDE the binary pattern of the numbers!
Say you want to repeat the first 100 terms. That means you want to calculate: fib(1) .. fin(101). It would occur at every intervals of 3 * 2**50 + i. That means:
fibonacci(3 * 2 ** 50 + i) & fibonacci(i) = fibonacci(i)
Where & means logical AND and fibonaccii is the function that calculates the xth number in the series.
Fibonacci will repeat inside the binary encoding from 3 * 2**n + k for 1 <= k <= 2n + 1.
If you want to find all the offsets in fibonacci where the first 10 terms occur, in binary, you simply calculate:
3 * 2^4 * k + 1. The first offsets would be [1, 49, 97, 145, 193, 241, …]. When k is 0, the first and initial set of the first 10 terms occur in fibonacci(1), fibonacci(2) .. fibonacci(10).
The next step would be fibonacci(49), fibonacci(50) .. fibonacci(59).
fibonacci(49) = 7778742049, in binary: 0b111001111101001100010111100100001
fibonacci(50) = 12586269025, in binary: 0b1011101110001100110011100101100001
fibonacci(51) = 20365011074, in binary: 0b10010111101110110010110100010000010
fibonacci(52) = 32951280099, in binary: 0b11110101100000011001010000111100011
fibonacci(53) = 53316291173, in binary: 0b110001101001111001100000101001100101
fibonacci(54) = 86267571272, in binary: 0b1010000010101111100101010110001001000
fibonacci(55) = 139583862445, in binary: 0b10000001111111110110001011011010101101
fibonacci(56) = 225851433717, in binary: 0b11010010010101110010110110001011110101
fibonacci(57) = 365435296162, in binary: 0b101010100010101101001000001100110100010
fibonacci(58) = 591286729879, in binary: 0b1000100110101011011011110111110010010111
fibonacci(59) = 956722026041, in binary: 0b1101111011000001000100111001011000111001
Look at the least significant bits! It's fibonacci, repeating inside itself!