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Collatz-type 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 |
Collatz-type problems with multiple divisors
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Creator | |
Publisher |
University of British Columbia
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Date Issued |
2009
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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.
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Extent |
453927 bytes
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File Format |
application/pdf
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Language |
eng
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Date Available |
2009-08-31
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0067665
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Program | |
Affiliation | |
Degree Grantor |
University of British Columbia
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Graduation Date |
2009-11
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Scholarly Level |
Graduate
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DSpace
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Rights
Attribution-NonCommercial-NoDerivatives 4.0 International