1. Introduction

To begin with, let us scrutinise the five reported GS1s discovered so far:

GS1(5): Prime(z) = {3,5}
GS1(7): Prime(z) ={3,5,7}
GS1(13): Prime(z)={3,5,7,11,13}
GS1(19}: Prime(z)= {3,5,7,11,13,17,19}
GS1(109}: Prime(z)={3,5,7,11,13,................,101,103,107,109}

Some common properties exist in the above full GS1s, viz., the first two integers are primes and the last two integers are also primes. Even GS1(5) satisfies this property but from GS1(7) to GS1(109), there are actually three primes at the beginning of these sequences. It is impossible to have three consecutive primes at the high ends because there are no further prime triplets above that of {3,5,7}. Another property connected with Prime(z) is that from GS1(7) onward, there will be increasing number of gaps in the prime sequence since some of the odd integers are composites. These gaps contribute to the loss of contiguities in Odd(z)^2 which might explain why beyond GS1(109), full Goldbach's sequences are so rare.

Here is the question posed: What are the conditions that will cause Odd(z)^2 to loose contiguity?

We conduct heuristic numerical investigations using an algebraic package. Instead of using Prime(z), it is more systematic and convenient to use Odd(z). This is because Odd(z) is a superset of Prime(z) being made up of the sum of primes and odd composites. Thus we can think of Odd(z) as the upperbound of Prime(z). By starting with a contiguous sequence of Odd(z) with an arbitrary number of terms, we can remove arbitrarily any single intermediate odd integer and compute Odd(z)^2 using an algebraic package. Then we can check the contiguity of the resultant even integer sequence. Table 1 summarises our findings for odd sequences from 3 terms to 9 terms starting from 3. Note that GS1(109) has not been investigated as it involves too many terms. This does not mean GS1(109) is not interesting. Finding out the reasons why there is this single isolated GS1 which breaks away from the first four GS1 would most likely give insights on how GS1(109) can contribute a full Goldbach's sequence. We will leave the investigations of GS1(109) to be reported in a future short paper.

Table 1 - Effects on contiguity of Odd(z)^2 when a single intermediate odd integer is
removed from Odd(z). Integer sets marked with asterisk * are noncontiguous.