Greatest Prime Factor
GPF 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Outline
1
Greatest Prime Factor
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
2
GPF Sequence Graphs
Graph Theory
GPF Sequence Graphs
Results
3
Conjectures and Empirical Evidence
Counting Connected Components
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result 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 its
factors are 1 and itself.
Any natural number, n, has a unique prime factorization
em for some natural number m and prime numbers
n = p1e1 . . . pm
p1 . . . pm .
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result 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 its
factors are 1 and itself.
Any natural number, n, has a unique prime factorization
em for some natural number m and prime numbers
n = p1e1 . . . pm
p1 . . . pm .
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result 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 its
factors are 1 and itself.
Any natural number, n, has a unique prime factorization
em for some natural number m and prime numbers
n = p1e1 . . . pm
p1 . . . pm .
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N − {1} → P
em
To compute the value of gpf (n), first consider n = p1e1 . . . pm
Then, gpf (n) = max{p1 , . . . , pm }.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N − {1} → P
em
To compute the value of gpf (n), first consider n = p1e1 . . . pm
Then, gpf (n) = max{p1 , . . . , pm }.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
gpf (n) : N − {1} → P
em
To compute the value of gpf (n), first consider n = p1e1 . . . pm
Then, gpf (n) = max{p1 , . . . , pm }.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined as
xn+1 = gpf (axn + b) where x0 is prime, and a, b ∈ N.
For example, let x0 = 2, a = 3, b = 1. Then
xn = 2, 7, 11, 17, 13, 5, 2, 7, 11, 17, 13, 5, 2, . . ..
Let x0 = 7, a = 5, b = 3. Then
xn = 7, 19, 7, 19, 7, 19, 7, 19, 7, 19, . . ..
Let x0 = 9, a = 2, b = 1. Then
xn = 9, 19, 13, 3, 7, 5, 11, 23, 47, 19, 13, 3, . . ..
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined as
xn+1 = gpf (axn + b) where x0 is prime, and a, b ∈ N.
For example, let x0 = 2, a = 3, b = 1. Then
xn = 2, 7, 11, 17, 13, 5, 2, 7, 11, 17, 13, 5, 2, . . ..
Let x0 = 7, a = 5, b = 3. Then
xn = 7, 19, 7, 19, 7, 19, 7, 19, 7, 19, . . ..
Let x0 = 9, a = 2, b = 1. Then
xn = 9, 19, 13, 3, 7, 5, 11, 23, 47, 19, 13, 3, . . ..
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined as
xn+1 = gpf (axn + b) where x0 is prime, and a, b ∈ N.
For example, let x0 = 2, a = 3, b = 1. Then
xn = 2, 7, 11, 17, 13, 5, 2, 7, 11, 17, 13, 5, 2, . . ..
Let x0 = 7, a = 5, b = 3. Then
xn = 7, 19, 7, 19, 7, 19, 7, 19, 7, 19, . . ..
Let x0 = 9, a = 2, b = 1. Then
xn = 9, 19, 13, 3, 7, 5, 11, 23, 47, 19, 13, 3, . . ..
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Sequence
A greatest prime factor sequence is defined as
xn+1 = gpf (axn + b) where x0 is prime, and a, b ∈ N.
For example, let x0 = 2, a = 3, b = 1. Then
xn = 2, 7, 11, 17, 13, 5, 2, 7, 11, 17, 13, 5, 2, . . ..
Let x0 = 7, a = 5, b = 3. Then
xn = 7, 19, 7, 19, 7, 19, 7, 19, 7, 19, . . ..
Let x0 = 9, a = 2, b = 1. Then
xn = 9, 19, 13, 3, 7, 5, 11, 23, 47, 19, 13, 3, . . ..
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
Greatest Prime Factor Function
gpf (3a + 1) for prime a
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelhoff conjectured that all gpf sequences are
ultimately 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) is
ultimately periodic
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelhoff conjectured that all gpf sequences are
ultimately 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) is
ultimately periodic
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Fundamental Theorem of Arithmetic
Greatest Prime Factor Sequences
Result on GPF Sequences
A Conjecture and Result on GPF Sequences
Caragiu and Sheckelhoff conjectured that all gpf sequences are
ultimately 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) is
ultimately periodic
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Some graph theory terms
A graph is a set, G = {V (G ), E (G )}, where V (G ) is a set of
vertices 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
GPF Sequence Graphs
We want to graphically represent the sequences.
Define U ax + b where a, b ∈ N, is the graph obtained by
taking graph unions between gpf sequences of all possible
initial sequences where the nodes are the prime numbers, and
edges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of all
of these sequences.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
GPF Sequence Graphs
We want to graphically represent the sequences.
Define U ax + b where a, b ∈ N, is the graph obtained by
taking graph unions between gpf sequences of all possible
initial sequences where the nodes are the prime numbers, and
edges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of all
of these sequences.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
GPF Sequence Graphs
We want to graphically represent the sequences.
Define U ax + b where a, b ∈ N, is the graph obtained by
taking graph unions between gpf sequences of all possible
initial sequences where the nodes are the prime numbers, and
edges are adjacent sequence members.
Graphs helps us to better consider the entire “universe” of all
of these sequences.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Some Examples
U3x+1
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Some Examples
U3x+4
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Some Examples
U2x+17
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Results
Theorem
For U ax + b and for any m, n ∈ N, any n-cycle A and any m-cycle
!
B, A B $= 0/ if and only if A = B.
Proof.
(→) Suppose that there exists an m-cycle, A and an n-cycle, B
!
!
such 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 know
that 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Results
Theorem
For U ax + b and for any m, n ∈ N, any n-cycle A and any m-cycle
!
B, A B $= 0/ if and only if A = B.
Proof.
(→) Suppose that there exists an m-cycle, A and an n-cycle, B
!
!
such 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 know
that 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Results
Theorem
For Uax+b , there does not exist a path between any disjoint cycles
A and B.
Proof.
Suppose an n-node path, a1 , . . . , an exists from A to B such that
a0 ∈ 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 have
gpf (a · a2 + b) = a1 , and likewise gpf (a · ai + b) = ai −1 for
i ≤ 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Graph Theory
GPF Sequence Graphs
Results
Results
Theorem
For Uax+b , there does not exist a path between any disjoint cycles
A and B.
Proof.
Suppose an n-node path, a1 , . . . , an exists from A to B such that
a0 ∈ 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 have
gpf (a · a2 + b) = a1 , and likewise gpf (a · ai + b) = ai −1 for
i ≤ 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 Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Counting Connected Components
Number of Connected Components?
Conjecture: Uax+a has only one connected component.
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Counting Connected Components
Connected Components
U23x+23
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Counting Connected Components
Connected Components
U6x+6
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Counting Connected Components
Thank you!
Tom Cuchta
GPF Sequence Graphs
Greatest Prime Factor
GPF Sequence Graphs
Conjectures and Empirical Evidence
Counting Connected Components
References I
Caragiu, M. and Scheckelhoff, L.
The Greatest Prime Factor and Related Sequences
JP 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., Mossinghoff, M.J.
Combinatorics and Graph Theory.
William Stein et al.
Sage Mathematics Software (Version 4.1), The Sage
Development Team, 2009,
http://www.sagemath.org/.
Tom Cuchta
GPF Sequence Graphs