Collatz Conjecture for 2^100000-1 Is True - Algorithms for Verifying Extremely Large Numbers

Abstract

Collatz conjecture (or 3x+1 problem) has been explored for about 80 years. By now the largest number that has been verified for Collatz conjecture is about 60 bits. In this paper, we propose new algorithms that can verify much greater numbers than known algorithms, i.e., about 100000 bits (30000 digits) on whether they can return 1 after 3*x+1 and x/2 computations. The proposed algorithms change numerical computation to bit computation (or logical computation), and change inner memory data representation to hard disk file manipulation. We make it possible to verify extremely large numbers without concerning computer memory upper-bound. The largest number verified for Collatz conjecture until now in the world is 2^100000-1, and this magnitude of extremely large numbers has never been verified. We discovered that this number can return to 1 after 481603 times of 3*x+1 computation, and 863323 times of x/2 computation. At last, we stress that some open problems may be used for the green mining in blockchain-based cryptocurrencies, e.g., BTC. We discuss the applicability of Collatz conjecture for the Proof of Work in Blockchain.