Open Collections

Abstract

The Collatz Conjecture hypothesizes that if a sequence of integers beginning with any positive integer t₀ is recursively defined so that t_j₊₁ = (t_j)/2 when t_j is even and t_j₊₁ = 3(t_j)+1 when t_j is odd, then there will be some j in the set of natural numbers such that t_j = 1. I propose a similar family of problems (which I call systems) involving a set of prime divisors {p₁,...,p_k} and a multiplier m, where the sequence is recursively defined so that t_j₊₁ = (t_j)/(p₁) if t_j is divisible by p₁, t_(j₊₁) = (t_j)/(p₂) if t_j is divisible by p₂ but not p₁, t_(j+1) = (t_j)/(p₃) if t_j is divisible by p₃ but not p₁ or p₂ etc., and if t_j is not divisible by any of the primes, then t_(j₊₁) = m(t_j)₊₁. Assuming the residues of the terms of these sequences behave randomly modulo p₁...p_k, I propose a multiplicative expectation and data to suggest that this is a reasonable model for these systems. If the expectation is less than 1, as in the case of the Collatz problem, then I hypothesize that any sequence will eventually result in some finite cycle. As well, if my model for these systems is accurate, then I prove that the inclusion of an increasing prime q to a fixed set of prime divisors will result in an effect that gradually diminishes for the multiplicative expectation of the system.