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