In Search Of XGS3-Primes
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~activweb/
Related URL-sites: http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 10/12/97. Revised:10/12 )
Abstract
The XGS-sequence described in a previous paper is based on the expansion of Prime(x)^2
and thus is a number sequence in the 2-dimensional sequence space [15]. For ease of
reference let it be called XGS2. By taking the cubic product of prime sequences, i.e.
Prime(x)^3, we establish XGS3 in 3-dimensional sequence space. XGS3-primes are even
rarer than XGS2-primes. The rule used for predicting the probable occurrences of XGS2-
primes cannot be applied to XGS3-primes. The only recognisable property common to
XGS2 and XGS3 seems to come from geometrical symmetries. Whereas XGS2 have 2-fold
symmetries, XGS3-has 2- and 3-fold symmetries. After investigating XGS3, there is
obviously a necessity to investigate higher ordered XGSs with the hope of finding common
properties amongst these XGS-groups.
1. Introduction
The XGS3-Primes are obtained by primality tests from the 3-dimensional or cubic XGS
sequence. Relevant definitions are given as follows:
Definiton: A 3-dimensional sequence number space is formed by the cubic product
Nat(x)*Nat(y)*Nat(z) where Nat(x), Nat(y), and Nat(z) are mutually orthogonal and the
product gerenates a discrete version of Euclidian 3-space.
XGS3-Sequence: Mathematically, XGS3:= Prime(x)*Prime(y)*Prime(z). This can be
collapsed to Prime(x)^3 as a line represented by the "cubic" diagonal in 3-dimensional
sequence space.
XGS3-Primes: These are found by primality tests on the numerator coefficients of XGS3-
sequence.
Here is a short example:
Primex:=sum(i/x^i*(isprime(i)-false)/(true-false),i=2..1000);
Primex :=
...3.........257....263....269.....43.....79.....83....37....331....499....503....223....293....673
---- +....+ ---- + ---- + ---- + --- + --- + --- + --- + ---- + ---- + ---- + ---- + ---- + ----
.....3.........257....263....269....43.....79.....83.....37....331....499....503....223....293....673
...x..........x........x........x........x........x.......x.......x......x.........x........x........x........x........x
+..............................................+...................................
....967....971....977....983....991....997....797....809
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- ...................................(1).
.....967....971....977....983....991....997....797....809
....x........x........x........x........x........x........x........x
XGS3:=sort(expand(Primex^3);
Only the odd numerator coefficients excluding those with last digits ending in 5s have been
collected into an XGS3-set for primality tests. The prime sequence used has an upperbound
of 1000. Therefore only the first one-third of XGS3-sequence is deterministic and primality
tested. It is seen that within the integer range from 2 to 96,000,000,000 there are only 7
XGS3-primes. There are more Mersenne's prime within this range since the 8th Mersenne's
prime of 2^31-1 = 2,147,483,647. XGS3-primes are shown in bold fonts in equation (2).
XGS3:=
{87 219 887 1419 7479 11703 85569 133107 227679
339651 357189 488411 1089033 2102319 2770023 2201373 3755937
3657477 2838249 4912989 3786259 6159537 7421883 17977257
15754269 15996357 23416947 18091493 29083647 41940537 34715367
29552501 45876579 35094799 60260211 40296231 70281897 66636549
116778909 75771711 133005597 126798399 89480367 148137723
154829949 179049657 128604881 204282183 198325833 232138149
208827537 176027201 193769523 280079829 337545933 332074929
346064637 358975809 521640009 581681889 559423311 440016937
878694237 819841077 582696207 930455427 823866369 625403097
954436557 1050961389 1262633199 1368091083 1399977711 1429570911
1594929567 1756180509 1665910389 1858195647 2006554347 1395942903
1423735593 1568519397 2587635441 2561643621 2884483053 2951391651
2155875751 3398304123 3030076389 3620307261 3442712667 2516051833
4028621061 4204417263 4561608369 3936338589 4690788921
4842748629 3296567873 5112980151 5775409167 4196313721 5983138617
4262880753 6374911677 6657282921 7520210529 6581846919
8340499029 7971983229 6361745863 9354736809 6618759027
10409208723 6793106541 11270860881 7755670371 12018117243
8387883571 11954167593 13337945967 14587866861 12783201123 15010915743
10301289259 10213951701 15459964089 11911945227 17961299181
12721172477 20317358781 13676104407 14731890977 22162378677 25324425843
17539763601 25855022271 25197355239 17887015613 17797809423
31752234063 30548328927 20909879349 37107450267 31343784411
35904934611 36699846123 25524337611 38847256989 42185743197
39819078879 37914387117 42314340153 30677737391 42860381991
30819404461 47970425907 46559214939 29249325477 50219966571
31857866487 49846065087 48663881763 36040974851 53508109827
38003494929 58076105019 60407229549 41822574543 65032047483
62178907191 65712118869 46810040073 65789550849 47733985421
70582728525 75734105373 78406100439 71835131109 52871411055
78782261007 87557465193 80845092471 53291408913 90664302711
59556190863 92780075343 80677647741 60227283093 63170187319
95097867537 64058820573 } ...............................................(2).
2. Prediction Of XGS3-Primes
Compared to XGS2, the occurrences of XGS3-primes are much less predictable than
the formers. A simple rule which predicts the probable occurrences of XGS2-primes exists and
is stated as follows:
Rule For XGS2-primes: Any summation of binary products of prime factors which does not
contain one term of the product of identical primes can never be a prime.
Note that there is no possibility of more than a single term with idetnical primes. Only a
primality test can determine whether the odd sum is an XGS-prime.
In order to find out the rules for predicting XGS3-primes, it is necessary to expand the
XGS3-sequence algebraically using equations (3) and (4):
Primex:=sort(sum(x(i)/x^i*(isprime(i)-false)/(true-false),i=2..50));
Primex :=
x(2)....x(3).....x(5)...x(7)....x(11)....x(13)....x(17)....x(19)...x(23)...x(29)...x(31)...x(37)
---- + ---- + ---- + ---- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + -----
.....2........3........5........7.........11........13........17........19........23.......29.......31.......37
...x.......x.........x........x..........x...........x..........x...........x.........x..........x.........x........x
..x(41)...x(43).....x(47)
+ ----- + ----- + ----- + ----- ....................................................(3).
.......41........43.........47
.....x..........x...........x
XGS3:=sort(expand(Primex^3));
A Partial listing is shown only for page economy.
.....................3...........2.........................2...........3.............2.....................................................2
...............x(2).......x(2) x(3)......x(2) x(3).......x(3).........x(2) x(5)........x(2) x(3) x(5).......x(2) x(7)
XGS3 := ----- + 3 ---------+ 3 ----------+ ----- + 3 ---------- + 6 -------------- + 3 ----------
.....................6..............7..................8.............9.................9.....................10....................11
..................x...............x..................x..............x.................x.....................x......................x
+....................................+..........................
.................................................2....................................3................................................2
......x(2) x(5) x(7).......x(2) x(11).......x(3) x(5) x(7)....x(5)........x(2) x(3) x(11).....x(2) x(7)
+ 6 --------------+ 3 ----------- + 6 --------------+ ----- + 6 ---------------+ 3 ----------
.............14........................15.....................15...............15...................16...................16
............x..........................x.......................x.................x.....................x.....................x
+................................................+...................
......................2.....................................................2......................................................2
......x(7) x(19).......x(11) x(17) x(19).....x(13) x(17)......x(13) x(17) x(19).....x(11) x(19)
+ 3 -----------+ 6 -----------------+ 3 ------------+ 6 -----------------+ 3 ------------ ....(4).
..............45.......................47.......................47.........................49........................49
.............x.........................x.........................x...........................x.........................x
Rules To Search For XGS3-Primes
From the XGS3-sequence expansions in equation (4), individual terms can be classified into
3 basic types with an example given for each type:
Basic Type 1: Product triplets with three identical prime factors. These always fall on the
diagonal line, i.e., the diagonal of opposite corners of a cube.
As shown in example 1 below, the first term is the type 1 term. All individual triplet product
terms are odd. The first term belongs to Basic Type 1 and the second term belongs to Basic
Type 2.
Example 1:
...............................................................3.............2
.........................................................x(3).........x(2) x(5)
XGS3 := ..................................... + ----- + 3 ---------- +....................
..............................................................9................9
............................................................x................x
Basic Type 2: Product triplets with two identical prime factors and one distinct prime factor.
These always fall on the normal dividers of the three sides of the isoceles triangular plane
normal to the diagonal line. These normal dividers intersect the diagonal line
normally and are coplanar with the normal plane.
As shown in example 1 above, the second term belongs to Basic Type 2. Two prime factors
are identical and one is distinct. These always fall on the normal dividers of the sides of the
isoceles triagular normal plane. There are 3 such dividers. That is why Type 2 always occur
in triplicates.
Basic Type 3: Product triplets with three distinct prime factors. These are always confined
to the isoceles triangular normal plane and always occur in image pairs about the three
normal dividers. Due to 3-fold symmetries, that is why these terms always occur in multiples
of 6.
Example 2:
.................................................2........................................3
..................................x(2) x(11)........x(3) x(5) x(7)......x(5)
........................... + 3 ----------- + 6 -------------- + ----- + .....................
..........................................15.....................15...............15
.........................................x.......................x.................x
The first term belongs to Type 2 and the third term belongs to Type 1 which have already
been described previously. The second term is a triplet product of distinct prime factors.
These always occur in image pairs in the normal plane symmetrical about the three normal
dividers. That is why these always occur in hexatuplets.
In XGS2, we have only 2-fold symmetries which makes predictions of the probable occurrences
of primes a possibility. In XGS3 we have both 3-fold and 2-fold symmetries and the sum of
off-diagonal terms can be even or odd. We
can thus make the following rules to assist the search for XGS3-primes:
Having defined these three classifiction of basic types, the following properties can be
elicited from them:
If the summation of terms give an odd value, then it is worthwhile doing a primality
test for primeness. It is possible to have multiples of Type 2 and Type 3 but there can
only be one term for Type 1. Table 1 summarises the rules:
Table 1 - Prediction Of Probable Occurrences Of XGS3-Primes
---------------------------------------------------------------------------------------------
....................Number of Basic Type Terms In Summations
---------------------------------------------------------------------------------------------
Basic Type 1......Basic type 2....Basic Type 3..Occurrence Of XGS3-Primes
(Odd)..................(Odd)................(Even).............Probable......Never
---------------------------------------------------------------------------------------------
......1......................................................................X
......1......................1......................................................................X
......1..............................................1......................X
......1......................2..............................................X
......1..............................................2......................X
..............................1..............................................X
......................................................1..............................................X
..............................1......................1......................X
..............................1......................2......................X
..............................2......................1..............................................X
..............................2......................2..............................................X
-----------------------------------------------------------------------------------------------
(Occurrences of more than one copy of Basic Type 1 means there is more
than one normal surface to the diagonal which is an impossibility in 3-space.
If there are odd multiples of Type 2, then the sum is odd but if there are
even multiples of Type 2 the sum is even. Both odd and even multiples of
Type 3 will return an even value. To use the above table, reduce Type 2 and
Type 3 to either 1 or 2 multiples before using this Table.)
3. Conclusions
XGS3-primes are much rarer than XGS2-primes. Within the integer range from 2 to
96,000,000,000 there are only seven such primes. At least within this finite range, XGS3-
primes are rarer than Mersenne's primes.
In XGS3-sequence, there exist 2-fold and 3-fold symmetries amongst the product terms.
From these symmetries a table of rules is setup which could help in narrowing down the
search for XGS3-primes. Nevertheless one still needs to apply primality tests on these
coefficients. Other than properties related to symmetries, there does not seem to be
any common property between XGS2 and XGS3. Most probably the common properties
could be grouped into two great divides depending whether Prime(x) is raised to even or
odd power. This picture will be completed by investigating higher ordered XGSs.
No attempt has been made to develop an open-form generating function predicting
the XGS3-sequence based on symmetry considerations. This is because the formulation
can get very complicated. In any case, it is still not possible to implement the
isprime function to do primality tests automatically within the formulation even if
one succeeds in developing it.
4. Reference:
[Comments: Papers included in this section are relevant for background readings
but not all are directly referenced in the main text. You can hyperlink to most of these
papers from within this reference section.
========================================================
1. Huen Y.K.:
A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994,
Vol.25, No.6, pp 913 - 920.
=======================================================
2. Huen Y.K.:
Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997,
Vol.28, No.3, 333-344.
=======================================================
3. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
========================================================
4. Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z) - by Huen Y.K. (date released : 20.6.97) (36.8 KBytes, 10*A4s).
========================================================
5. Lemmata, Corollaries, And
Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).
========================================================
6. Improved Formulations For Comp(z)
and Prime(z)
- by Huen Y.K. (date released : 16.9.97) (17 KBytes ).
========================================================
7. Detecting False Reports
in Primality Tests By The Oddcomp(z) Method.
- by Huen Y.K. (date released : 18.9.97, Revised 20/9) (26 KBytes ).
========================================================
8. The Throwing Power Of
Oddcomp(z).
- by Huen Y.K. (date released : 24.9.97 ) (15 Kbytes).
========================================================
9. Sequence Algebraic
Approach To Prime Number Theorem
- by Huen Y.K. (date released : 28.9.97 ) (21 Kbytes).
========================================================
10. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
========================================================
11. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
========================================================
12. In Search Of Counter-
Examples In Maple's Isprime Function.
- by Huen Y.K. (date released : 4.10.97 ) (18 Kbytes).
========================================================
13. A Sequence Algebraist's
View Of Lehmann's Primality Test
- by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).
========================================================
14. On Odd(z), Oddcomp(z),
Seq1(z) and Seq2(z)
- by Huen Y.K. (date released : 10.10.97 ) (17 Kbytes).
========================================================
15.In Search Of Primes From
The XGS Sequence
- by Huen Y.K. (date released : 10.12.97 ) (26 Kbytes).
=========================================================
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