The Strange World of the
Hausdorff Metric Geometry
XV. SPACK Numbers
The investigations in 2005 REU of the number
of sets at each location between two sets A and B has led us to
define special types of integers we call SPACK numbers that provide a new
partition of the set of natural numbers.
Definition: A positive integer k is a SPACK-n number if
there exists a configuration of two sets A
and B in Rn having
exactly k elements at each location
between A and B, and no such configuration exists in Rn-1. If no such configuration exists in any
dimension, then k is called a SPACK-0
number. If a SPACK-n number p is prime, then
we call p a SPACK-n prime.
In case you are wondering, SPACK stands for Steve-Patrick-Alex-Chantel-Kris,
after the originators of the idea.
The only configurations which can be realized
on the real number line R1
are single or multiple m-strings. The multiplicative property of configurations
demonstrates that if we have more than one string on the real line, the number
of elements at each location will be a composite number. So SPACK-1 numbers are only those integers
that can be written as Fibonacci numbers or as a product of Fibonacci
numbers. We can easily see that the only
SPACK-1 primes will be those numbers that are Fibonacci primes. It is widely suspected but yet to be proven
that there are infinitely many SPACK-1 primes [12].
Furthermore, we can realize every Lucas
number in R2, so it is
also believed that there infinitely many SPACK-2 primes. As we have seen, the
prime 19 is a SPACK-0 number. In the 2007 REU, Katrina Honigs connected finite
configurations to bipartite graphs and showed that 37 is also a SPACK-0 number
[11]. Alex Zupan has conjectures about other SPACK-0 numbers (in an upcoming
paper, I hope) and Dan Schultheis (REU 2006) has shown using a brute force
approach that 57 is a SPACK-3 number (the only confirmed one we know). The list of known SPACK numbers to date is:
·
SPACK-0
numbers: 19, 37
·
SPACK-1
numbers: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, ...
·
SPACK-1
primes: 2, 3, 5, 13, 89, 233, 1597, 28657, ...
·
SPACK-2
numbers: 7, 11, 14, 17, 22, 23, 28, 29, 31, 33, 35, 38, 43, 44, ...
·
SPACK-2
primes: 7, 11, 17, 23, 29, 31, 43, 47, ...
·
SPACK-3
numbers: 57 and possibly 79
·
SPACK-4+
numbers: unknown