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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  
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

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Type  
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  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
200911

Campus  
Scholarly Level 
Graduate

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Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivatives 4.0 International