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Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
GPF Sequence Graphs
Tom Cuchta
Department of Mathematics
Ohio MAA Meeting, 2009
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Outline
1 Greatest Prime FactorFundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
2 GPF Sequence GraphsGraph TheoryGPF Sequence GraphsResults
3 Conjectures and Empirical EvidenceCounting Connected Components
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Fundamental Theorem of Arithmetic
A natural number n is an element of the set N = {1,2,3, . . .}.
A prime number p ! P is a natural number such that itsfactors are 1 and itself.
Any natural number, n, has a unique prime factorizationn = pe1
1 . . .pem
m for some natural number m and prime numbersp1 . . .pm.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Fundamental Theorem of Arithmetic
A natural number n is an element of the set N = {1,2,3, . . .}.
A prime number p ! P is a natural number such that itsfactors are 1 and itself.
Any natural number, n, has a unique prime factorizationn = pe1
1 . . .pem
m for some natural number m and prime numbersp1 . . .pm.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Fundamental Theorem of Arithmetic
A natural number n is an element of the set N = {1,2,3, . . .}.
A prime number p ! P is a natural number such that itsfactors are 1 and itself.
Any natural number, n, has a unique prime factorizationn = pe1
1 . . .pem
m for some natural number m and prime numbersp1 . . .pm.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N"{1}# P
To compute the value of gpf (n), first consider n = pe11 . . .pem
m
Then, gpf (n) = max{p1, . . . ,pm}.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N"{1}# P
To compute the value of gpf (n), first consider n = pe11 . . .pem
m
Then, gpf (n) = max{p1, . . . ,pm}.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N"{1}# P
To compute the value of gpf (n), first consider n = pe11 . . .pem
m
Then, gpf (n) = max{p1, . . . ,pm}.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined asxn+1 = gpf (axn +b) where x0 is prime, and a,b ! N.
For example, let x0 = 2, a = 3, b = 1. Thenxn = 2,7,11,17,13,5,2,7,11,17,13,5,2, . . ..
Let x0 = 7, a = 5, b = 3. Thenxn = 7,19,7,19,7,19,7,19,7,19, . . ..
Let x0 = 9, a = 2, b = 1. Thenxn = 9,19,13,3,7,5,11,23,47,19,13,3, . . ..
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined asxn+1 = gpf (axn +b) where x0 is prime, and a,b ! N.
For example, let x0 = 2, a = 3, b = 1. Thenxn = 2,7,11,17,13,5,2,7,11,17,13,5,2, . . ..
Let x0 = 7, a = 5, b = 3. Thenxn = 7,19,7,19,7,19,7,19,7,19, . . ..
Let x0 = 9, a = 2, b = 1. Thenxn = 9,19,13,3,7,5,11,23,47,19,13,3, . . ..
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined asxn+1 = gpf (axn +b) where x0 is prime, and a,b ! N.
For example, let x0 = 2, a = 3, b = 1. Thenxn = 2,7,11,17,13,5,2,7,11,17,13,5,2, . . ..
Let x0 = 7, a = 5, b = 3. Thenxn = 7,19,7,19,7,19,7,19,7,19, . . ..
Let x0 = 9, a = 2, b = 1. Thenxn = 9,19,13,3,7,5,11,23,47,19,13,3, . . ..
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined asxn+1 = gpf (axn +b) where x0 is prime, and a,b ! N.
For example, let x0 = 2, a = 3, b = 1. Thenxn = 2,7,11,17,13,5,2,7,11,17,13,5,2, . . ..
Let x0 = 7, a = 5, b = 3. Thenxn = 7,19,7,19,7,19,7,19,7,19, . . ..
Let x0 = 9, a = 2, b = 1. Thenxn = 9,19,13,3,7,5,11,23,47,19,13,3, . . ..
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
Greatest Prime Factor Function
gpf (3a+1) for prime a
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelho! conjectured that all gpf sequences areultimately periodic.
They showed it to be true for a particular family of functions.
Theorem
If a = 1 and b ! N, then if x0 is prime, xn+1 = gpf (xn +b) isultimately periodic
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelho! conjectured that all gpf sequences areultimately periodic.
They showed it to be true for a particular family of functions.
Theorem
If a = 1 and b ! N, then if x0 is prime, xn+1 = gpf (xn +b) isultimately periodic
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of ArithmeticGreatest Prime Factor SequencesResult on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelho! conjectured that all gpf sequences areultimately periodic.
They showed it to be true for a particular family of functions.
Theorem
If a = 1 and b ! N, then if x0 is prime, xn+1 = gpf (xn +b) isultimately periodic
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Some graph theory terms
A graph is a set, G = {V (G ),E (G )}, where V (G ) is a set ofvertices and E (G ) is a set of edges between those vertices.
A path in a graph is a sequence of edges, a0,a1, . . . ,an such that(ai ,ai+1) is a valid edge.
A cycle in a graph is a path, a0, . . . ,an such that a0 = an.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
GPF Sequence Graphs
We want to graphically represent the sequences.
Define Uax +b where a,b ! N, is the graph obtained bytaking graph unions between gpf sequences of all possibleinitial sequences where the nodes are the prime numbers, andedges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of allof these sequences.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
GPF Sequence Graphs
We want to graphically represent the sequences.
Define Uax +b where a,b ! N, is the graph obtained bytaking graph unions between gpf sequences of all possibleinitial sequences where the nodes are the prime numbers, andedges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of allof these sequences.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
GPF Sequence Graphs
We want to graphically represent the sequences.
Define Uax +b where a,b ! N, is the graph obtained bytaking graph unions between gpf sequences of all possibleinitial sequences where the nodes are the prime numbers, andedges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of allof these sequences.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Some Examples
U3x+1
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Some Examples
U3x+4
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Some Examples
U2x+17
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Results
Theorem
For Uax +b and for any m,n ! N, any n-cycle A and any m-cycleB, A
!B $= /0 if and only if A = B.
Proof.
(#) Suppose that there exists an m-cycle, A and an n-cycle, Bsuch that A
!B $= /0. Then, there exists a node, t ! A
!B . Since
x ! A, we know that gpf (at +b) ! A and since x ! B , we knowthat gpf (at +b) ! B . Thus, gpf (at +b) is also in the intersection.It follows inductively that A = B if A
!B $= /0.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Results
Theorem
For Uax +b and for any m,n ! N, any n-cycle A and any m-cycleB, A
!B $= /0 if and only if A = B.
Proof.
(#) Suppose that there exists an m-cycle, A and an n-cycle, Bsuch that A
!B $= /0. Then, there exists a node, t ! A
!B . Since
x ! A, we know that gpf (at +b) ! A and since x ! B , we knowthat gpf (at +b) ! B . Thus, gpf (at +b) is also in the intersection.It follows inductively that A = B if A
!B $= /0.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Results
Theorem
For Uax+b, there does not exist a path between any disjoint cyclesA and B.
Proof.
Suppose an n-node path, a1, . . . ,an exists from A to B such thata0 ! A and other ai $! A and an ! B with other ai $! B . Since a0
attached to a1, gpf (a ·a1 +b) = a0. Then, we must havegpf (a ·a2 +b) = a1, and likewise gpf (a ·ai +b) = ai"1 fori % 1 < n. Consider gpf (a ·an +b). To preserve the path,gpf (a ·an +b) = an"1, but an"1 $! B . Contradiction!
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical Evidence
Graph TheoryGPF Sequence GraphsResults
Results
Theorem
For Uax+b, there does not exist a path between any disjoint cyclesA and B.
Proof.
Suppose an n-node path, a1, . . . ,an exists from A to B such thata0 ! A and other ai $! A and an ! B with other ai $! B . Since a0
attached to a1, gpf (a ·a1 +b) = a0. Then, we must havegpf (a ·a2 +b) = a1, and likewise gpf (a ·ai +b) = ai"1 fori % 1 < n. Consider gpf (a ·an +b). To preserve the path,gpf (a ·an +b) = an"1, but an"1 $! B . Contradiction!
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical EvidenceCounting Connected Components
Number of Connected Components?
Conjecture: Uax+a has only one connected component.
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical EvidenceCounting Connected Components
Connected Components
U23x+23
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical EvidenceCounting Connected Components
Connected Components
U6x+6
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical EvidenceCounting Connected Components
Thank you!
Tom Cuchta GPF Sequence Graphs
Greatest Prime FactorGPF Sequence Graphs
Conjectures and Empirical EvidenceCounting Connected Components
References I
Caragiu, M. and Scheckelho!, L.The Greatest Prime Factor and Related SequencesJP J. of Algebra, Number Theory and Appl. 6 (2006) , 403 -409.
Guido van Rossum.Python Programming Language.http://python.org/.
Harris, J. M., Hirst J. L., Mossingho!, M.J.Combinatorics and Graph Theory.
William Stein et al.Sage Mathematics Software (Version 4.1), The SageDevelopment Team, 2009,http://www.sagemath.org/.
Tom Cuchta GPF Sequence Graphs