An Extension of the Collatz Conjecture modulo $2^p+2^q$

Abstract

In this paper, we will introduce an extension to the Collatz's conjecture. This conjecture may be seen as a general conjecture that unifies the Collatz one together with many other similar conjectures. For instance, we propose our new conjecture modulo $10$ which may be stated as follows. Starting from any positive integer, if it is a multiple of $10$ then divide it by 10, otherwise, multiply it by $12$, add $8$ times its last digit and divide the result by $10$. Repeat the process infinitely. Regardless the starting number, the process eventually reaches $4$ after a finite number of iterations. The genaral conjecture studied here will encompasse the classical Collatz conjecture togher with our proposed one modulo $10$.