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Collatztype problems with multiple divisors Gunn, Keira
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.
Item Metadata
Title  Collatztype problems with multiple divisors 
Creator  Gunn, Keira 
Publisher  University of British Columbia 
Date Issued  2009 
Description 
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.

Extent  453927 bytes 
Genre  Thesis/Dissertation 
Type  Text 
File Format  application/pdf 
Language  eng 
Date Available  20090831 
Provider  Vancouver : University of British Columbia Library 
Rights  AttributionNonCommercialNoDerivatives 4.0 International 
DOI  10.14288/1.0067665 
URI  
Degree  Master of Science  MSc 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Graduation Date  200911 
Campus  UBCV 
Scholarly Level  Graduate 
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Aggregated Source Repository  DSpace 
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AttributionNonCommercialNoDerivatives 4.0 International