The Strange World of the Hausdorff Metric Geometry

 

XIV. 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 [11].

 

Furthermore, we can realize every Lucas number in R2, so it is also believed that there infinitely many SPACK-2 primes. To date 19 is the only confirmed SPACK-0 number. At this time we conjecture that 37 may be a SPACK-0 number and that 57 and 79 may be SPACK-3 numbers.  The list of known SPACK numbers to date is:

 

·       SPACK-0 numbers: 19 (possibly 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: unknown, but 57 and 79 are candidates

·       SPACK-4+ numbers: unknown