What Is So Special About GS1(109):0?


by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg

(A short communication - 1st released: 16/8/97, revised:17/8,18/8)


Abstract

Goldbach's sequence has been defined by the author as Goldbach(z)=Normc(Prime(z)^2). The search for full Goldbach's sequences has been going on in this URL site since 1/2/97. If Prime(z) is confined to odd primes, then a full Goldbach's sequence is simply a contiguous even integer sequence starting from 6. If GS1(3) is excluded as a trivial case, up to now only five full Goldbach's sequences have been found, viz., GS1(5), GS1(7), GS1(13), GS1(19) and GS1(109). Search by Vincent Celier shows the absence of any more full GS1 from above 109 to below 1000_000_000. There is no proof that the number of full GS1 is finite. Amongst these five GS1s, some common properties on the first four have been reported in a previous paper [17]. The most intriguing GS1 is the fifth one which is also at present the last one. Whilst it shares some common properties with the other four, it is an unlikely candidate for a full GS1 in view of its length. There exists many gaps between primes so that the chance of generating a full GS1 is quite slim. Yet what is happenning here? That is what the author proposes to find out. Here are his prelminary findings.


1. Introduction

Currently only five full Goldbach's sequences, i.e., GS1, have been found as listed below:

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 is 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. Whilst some understanding on how the first four GS1 arise was obtained in a previous paper [17], it is the last one, i.e. GS1(109):0 which is most intriguing. In view of its length there exist many gaps between primes in the sequence. Yet how does it manage to generate a full Goldbach's sequence? This is what the author intends to find out.

The question asked is: Why is GS1(109) a full Goldbach's sequence?

Again 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) as far as odd integers are concerned. 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. Without computers, this will be quite a tedious chore. Fortunately, results can be obtained quickly using an algebraic package. Table 1 summarises the findings on GS1(109) by replacing it with Odd(z) and subsquently by Prime(z) ranging from 3 to 109.

The Maple line used to generate Odd(z) without the truncation order term is shown below:
Odd(z):=sort(series(1/(z*(z^2-1))-1/(z^110*(z^2-1)),z=infinity,110)-O(1/z^111));
For brevity, the odd integer sequence generated by Odd(z) is written in integer set notation as shown below and any odd integer removed will be replaced by an "X" within the set.

Odd(z):= {3 5 7 9 11 13 15 17 19 21 23 25 27 29
31 33 35 37 39 41 43 45 47 49 51 53 55 56 57 58
59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
106 107 108 109}

Table 1 - Effect on contiguity of Odd(z)^2 when a single intermediate odd integer is removed from Odd(z). Integer sets marked with asterisk * are noncontiguous.
Odd(z) .....................................................Odd(z)^2
Odd(z):= {3 5 7 9 11 13 15 17 19 21 23 25 27 29
31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93
95 97 99 101 103 105 107 109}

Odd(z)^2:={6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218}

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Odd(z):= {3 X 7 9 11 13 15 17 19 21 23 25 27 29
31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93
95 97 99 101 103 105 107 109}

Odd(z)^2:= {6 .. 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72
74 76 78 80 82 84 86 88 90 92 94 96 98 100 102
104 106 108 110 112 114 116 118 120 122 124 126 128
130 132 134 136 138 140 142 144 146 148 150 152 154
156 158 160 162 164 166 168 170 172 174 176 178 180
182 184 186 188 190 192 194 196 198 200 202 204 206
208 210 212 214 216 218}*

-----------------------------------------------------------------------------------------------------------------------------

Odd(z):= {3 5 7 9 11 13 15 17 19 21 23 25 27 29
31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93
95 97 99 101 103 105 X 109}

Odd(z)^2:= (6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 .. 218}*
-------------------------------------------------------------------------------------------------------------------------
Odd(z):= {3 5 X 9 11 13 15 17 19 21 23 25 27 29
31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61
63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93
95 97 99 101 103 105 107 109}

Odd(z)^2 is a full sequence.
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From this point, only Prime(z) sequence is used instead of Odd(z) sequence.

The objective is to study the effect of removing isolated primes which are marked as X in the prime sets. The number of removals is systematically increased until all the isolated primes are elminated. It can be seen that removing isolated primes does not cause loss of contiguity in Odd(z)^2 in GS1(109).

Prime(z):= { 3 5 7 11 13 17 19 23 29 31 37 41 43 47
51 53 59 61 67 71 73 77 79 83 89 91 95 97 101 103 107 109 }

Prime(z)^2:= { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
---------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 37 41 43 47
51 53 59 61 67 71 73 77 79 83 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
--------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 47
51 53 59 61 67 71 73 77 79 83 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
----------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 67 71 73 77 79 83 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
--------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 83 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
---------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
-------------------------------------------------------------------------------------------------------------------------
After completing the removal of isolated primes, we now systematically test the removal of one member of a twimprime pair other than the first prime triplet and the last twinprime in GS1(109). These removals are marked by "X".

Prime(z):= { 3 5 7 "X" 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 .. 30 32 34 36 38 42
44 46 48 50 54 56 58 60 62 64 66 68 70 72 74 76
78 80 82 84 86 88 90 92 94 96 98 100 102 104 106
108 110 112 114 116 118 120 122 124 126 128 130 132
134 136 138 140 142 144 146 148 150 152 154 156 158
160 162 164 166 168 170 172 174 176 178 180 182 184
186 188 190 192 194 196 198 200 202 204 206 208 210
212 214 216 218 }* --------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 "X" 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 97 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
-----------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 97 101 "X" 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 .. 214 216 218 }*
---------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 97 "X" 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 .. 210 212 214 216 218 }*
-------------------------------------------------------------------------------------------------------------------------
Prime(z):= { 3 5 7 11 13 17 19 X 29 31 X 41 43 X
51 53 59 61 X 71 73 77 79 X 89 91 95 "X" 101 103 107 109 }

Prime(z)^2 := { 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38
40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
72 74 76 78 80 82 84 86 88 90 92 94 96 98 100
102 104 106 108 110 112 114 116 118 120 122 124 126
128 130 132 134 136 138 140 142 144 146 148 150 152
154 156 158 160 162 164 166 168 170 172 174 176 178
180 182 184 186 188 190 192 194 196 198 200 202 204
206 208 210 212 214 216 218 }
--------------------------------------------------------------------------------------------------------------------------


3. Conclusions

GS1(109) is unusual because it has the densest populations of twinprimes for its length. It has already been observed that all full Goldbach's sequences are capped by twinprimes at both end. For GS1(109) the isolated primes are all located quite far in towards the centre of the sequence where they do not cause loss of contiguities. In GS1(109) there is another pair of twinprime in {101,103) just below the capping pair of {107,109). This explains why this particular sequence is quite immuned to the removals of isolated primes without loosing contiguity. Moving beyond GS1(109), it is most likely that the density of twinpimes starts decreaing which is partly caused by the decline in increase of primes. This is further aggravated by the larger gaps between primes. Vincent Celier has already shown by numerical investigation that below 1000_000_000 there is no further full Goldbach's sequence. Most probably, but there is no proof, that there will be no more full sequence above GS1(109). It will be a challenge to mathematicians to find a rigorous proof for it. This is currently an unsolved problem.

Note that the observations were made without thorough investigations of higher sequences. The tentative conclusions are that twinprimes play an important part in the contiguity of the first five GS1s but further up, the most likely reason for the rarity of GS1 is because of the growing gaps between primes. The common characteristic amongst the first five GS1 is that they are all capped by a prime triplet at the low end and a twinprime at the high end. This is the minimum condition for a full GS1. Considerable saving in computing resources could be effected by confining the search for GS1 amongst sequences which share this characteristics. Having two closely spaced pairs of twinprimes at the high end also helps. Density of twinprimes in the sequence also help. The known properties of the prime sequence are unfavourable to the existence of more full Goldbach's sequences above GS1(109). The author's conjecture is that there will be no more GS1 above GS1(109) but this can only be validated by a rigorous mathematical proof.


3 References
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