In Search Of Primes From The XGS Sequence
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: 8/12/97. Revised 8/12 )
Abstract
The Extended Goldbachs's Sequence (XGS) can only be written in the open form but
it is computationally more efficient than the closed form. This is done by including in
the numerators the binary
products of prime factors. The Extended Goldbach's Sequence is based on the
number sequence formed by these numerator coefficients instead of the indices
belonging to the order-variables in the denominators. The occurrences of primes
in this sequence is quite rare but not as rare as Mersenne's primes. Within the
bound of integer range from 2 to 434,000,000 only 31 such primes are found. A
necessary but not sufficient condition for the occurrence of an XGS prime is that
the numerator coefficient must be the sum of binary products of prime factors which
consists of a square of identical primes and an even number of products of
distinct primes which occur in image pairs. Geometrically, this means that the
summation must come from points with prime coordinates which occur along a
normal to the diagonal in the 2-dimensional sequence number space. Just like
in Mersenne's primes, XGS primes have to be picked out by primality tests.
1. Introduction
As a preliminary, some terms are defined as follows:
Definiton: A 2-dimensional sequence number space is formed by the binary product of
Nat(x)*Nat(y) where Nat(x) is orthogonal to Nat(y) and both represent discrete 1-dimensional
cartesian axes. It is simply a discrete version of Euclidian space.
XGS Sequence: The is a sequence formed by the product of two identical
prime sequences defined by: XGS(x,y) := Prime(x)*Prime(y). Effectively, it is a 2-
dimensional number sequence but when x=y, this collapses into the diagonal in the
2-dimensional number space as a 1-dimensional number line.
XGS Primes: These are primes along the XGS sequence obtained by
primality tests.
A necessary but insufficient geometrical condition for the occurrence of a XGS prime
is that the numerator coefficients in the XGS-sequence must be formed by the summation
of all binary products of prime factors which occur in the normal to the diagonal in the
2-dimensional sequence space. Amongst these binary products there must be one product
of an identical prime pair and an even number of products of distinct prime factors.
Here is a short example:
Primex:=sum(x(i)/x^i*(isprime(i)-false)/(true-false),i=2..11);
.................x(2)...x(3)....x(5)....x(7)....x(11)
Primex := ---- + ---- + ---- + ---- + ----- ............................(1).
.....................2.......3.........5........7.......11
..................x.......x..........x.........x........x
Primey:=sum(y(i)/y^i*(isprime(i)-false)/(true-false),i=2..11);
.................y(2)...y(3)....y(5)....y(7)....y(11)
Primey := ---- + ---- + ---- + ---- + ----- ...........................(2).
.....................2........3.......5........7.........11
...................y........y.......y........y.........y
Prime2:=sort(expand(Primex*Primey));
.......................x(3) y(11)...x(5) y(2)...x(5) y(3)...x(5) y(5).....x(5) y(7)...x(5) y(11)....x(7) y(2)
Prime2:=sort(---------- + --------- + --------- + --------- + --------- + ---------- + ---------
...........................3 11.............5 2............5 3.............5 5...........5 7............5 11............7 2
.........................x y...............x y.............x y..............x y............x y.............x y..............x y
...x(7) y(3)....x(7) y(5).....x(7) y(7)....x(7) y(11)....x(11) y(2)....x(11) y(3)....x(11) y(5)
+ --------- + --------- + --------- + ---------- + ---------- + ---------- + ----------
.........7 3............7 5..............7 7............7 11...........11 2............11 3............11 5
.......x y............x y..............x y..............x y..............x y...............x y...............x y
...x(11) y(7)....x(11) y(11)....x(2) y(2)....x(2) y(3)....x(2) y(5)....x(2) y(7)....x(2) y(11)
+ ---------- + ----------- + --------- + --------- + --------- + --------- + ----------
........11 7.............11 11.............2 2.............2 3.............2 5............2 7............2 11
.......x y...............x...y...............x y..............x y.............x y.............x y.............x y
...x(3) y(2)....x(3) y(3)....x(3) y(5)....x(3) y(7)
+ --------- + --------- + --------- + --------- ) ...............................................(3).
.........3 2.............3 3.............3 5...........3 7
.......x y.............x y.............x y............x y
Prime2 in equation(3) is a 2-dimensional number sequence. The binary products in the
denominators displays the individual prime factors and the order-variables x and y trace their
origins. Furthermore because of the exponential forms, the prime factors will be added in the
denominators but multiplied in the numerators. The summation of prime factors reminds one
of Goldbach's sequence but here these have no significance other than 2-dimensional
ordering. The modifications to the numerators turn this into a XGS-sequence.
Furthermore, if we put y = x, the 2-dimensional sequence equation (3) can be
collapsed into a 1-dimensional sequence where the ordering now can be interpreted as that of
a number sequence along the diagonal of the 2-dimensional sequence map with all off-
diagonal binary products mapped normally onto it. In fact all terms with the same sum of
indices in the denominators will come under the same normal. The binary product in the
numerators are unassigned array elements which could be used to delay numerical
computations. Normally we assign x(i) = i so that even before numerical assignments, we
already know the values of these prime factors.
Numerical Evaluation Of Equation (3):
This is done by putting y=x and x(i) = i for i = 2..11. The resultant sequence is shown in
equation (4).
Primex:=sum(i/x^i*(isprime(i)-false)/(true-false),i=2..11);
..................2........3.........5.......7.......11
Primex := ---- + ---- + ---- + ---- + --- ....................................(4).
....................2........3.........5.......7......11
..................x........x.........x.......x........x
Prime2:=sort(expand(Primex*Primex));
..................4.......12.......9........20......30......28......67.....70....44...115...110...154...121
Prime2 := ---- + ---- + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- ........(5).
....................4.......5.........6.........7........8........9.......10.....12....13.....14.....16.....18.....22
..................x........x.........x.........x........x........x.........x.......x......x.......x........x......x.......x
Amongst the numerators in equation (5), only 67 is a XGS prime where the
contributing terms can be traced from equation (3) as:
67 = 5*5 + 2*7*3 .......................................(6).
It can be seen from the above equation that the first term is the square of identical primes
and the second term will always be even in view of the presence of a prime factor 2. This is
contributed by the fact that the binary product 3*7 and 7*3 form an image pair one above
and one below the diagonal and the normal links all off-diagonal terms through the identical
prime pair which is always situated in the diagonal. There is no guarantee that the number
of binary terms increases with the magnitudes of the values of the numerator coefficients but
generally the number of binary terms are minimum at the two extremes of equation (5) and
maximum near the middle of the sequence. Furthermore, just like in the Goldbach's
sequence, the upper half of the sequence in equation (5) may not be fully filled. In other
words, when the length of the prime sequence is further increased, you might find some
missing terms surfacing. Therefore, in any product sequence, we are never sure we have
got all the terms with the exception of those in the lower half of the sequence which are fully
filled.
2. Known XGS Primes
Adopting the method described from equations (1) to (5), a large problem is computated
using a prime sequence upperbounded by 10000. The resultant binary product sequence will
be upperbounded by 19999. Only the lower half of the terms are primality
tested for XGS-primes. The numerator coefficients in the upper half of the sequence are
not "fully filled" and
will give erroneous answers to primality tests.
Table of the first 31 Industrial Grade QPs.
(the denominator terms are also included).
Found by Huen Y.K. (6.12.97)
------------------------------------------------------
67/z^10
391073/z^346
424811/z^382
2611997/z^694
2704549/z^706
3205999/z^802
4869817/z^974
5822963/z^1006
7556141/z^1154
8287141/z^1202
18410429/z^1522
23064389/z^1658
38727709/z^1954
42058241/z^2038
43405309/z^2174
46794731z^2182
66344797/z^2462
81010201/z^2554
75081073/z^2614
94027949/z^2798
28908313/z^2974
108703697/z^3086
128848373/z^3194
158211967/z^3334
155381599/z^3554
180596973/z^3566
243678751/z^3974
213575741/z^4922
341742833/z^4474
304443119/z^4478
433119287/z^4894
.....................
------------------------------------------------------
3. Probabilistic Generating Function For XGS-Sequence
The probablilistic generating function for XGS-sequence is given by equations (7)
to (9). The
generating function will display binary products of prime factors for off-diagonal terms in x1
and diagonal terms in x2. The sum XGS=x1+x2 will generate the XGS-sequence starting from 4.
This will have to be tested separately.
The Maple lines predict the occurrences of XGS primes
correctly up to half the range, i.e., within the first 500 terms. One could obtain the algebraic expression by fully
expanding the Maple lines but it is not useful as it is too complicated. It is far better
to study the Maple lines instead.
x1:=sort(2*sum(sum((i+2*j)*(i-2*j)/z^(2*i)*(isprime(i+2*j)-false)*(isprime(i-2*j)-false)/(true
-false)^2,i=2..500),j=1..500/2));
x1 :=
42....66.....284...284...404..1182..982...434...2680..1654..3024..5354..2958..1988..8724
--- + --- + --- + --- + --- + ---- + --- + --- + ---- + ---- + ---- + ---- + ---- + ---- + ----
..10....14......18......22....26.....30.....34.....38......42......46......50......54......58......62......66
.z......z........z........z.......z........z........z.......z........z........z..........z........z........z.........z........z
+............................+...........
309516..356202..664370..361144..482706..760568..343170..267054..749460..590084
+ ------ + ------ + ------ + ------ .+ ------ .+ ------ .+ ------. + ------. + ------ .+ ------
.....334.........338........342.......346.......350.........354.......358........362........366.........370
....z.............z............z............z............z.............z............z.............z.............z.............z
453428..1023052..388330..414566..1342350..463340..267132..839666..661610..675122
+ ------ + ------- + ------ .+ ------. + ------- .+ ------ .+ ------ .+ ------ .+ ------ .+ ------
.....374.........378........382.........386........390..........394.........398........402........406........410
....z.............z.............z.............z.............z...............z.............z............z............z.............z
+............................+................................(7).
x2:=sort(series(sum(i*i/z^(i+i),i=2..500),z=infinity,500));
..............1..........4.........9......16......25.....36.....49....64......81...100...121...144...169...196...225
x2 := O(----) + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.............498.........4.........6........8......10......12....14....16......18....20....22......24.....26.....28.....30
.............z...........z..........z........z........z........z.......z.......z........z......z.......z........z.......z.........z......z
28561...28900..29241..29584..29929..30276..30625..30976..31329..31684..32041..32400
+ ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + -----
....338......340......342......344......346......348......350......352......354......356......358.......360
...z..........z..........z...........z..........z...........z...........z..........z...........z...........z..........z...........z
32761..33124..33489..33856..34225..34596..34969..35344..35721..36100..36481..36864
+ ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- + -----
....362......364......366......368......370......372......374......376......378......380......382.....384
....z..........z..........z...........z..........z...........z...........z..........z..........z............z..........z..........z
..............................................................................................(8).
XGS:=sort(x1+x2);
As predicted, only terms in the first half the XGS-sequence are correct. Thus the XGS
primes 67, 391073 and 424811 are correct when compared to those entered in Table 1.
Beyond
the midpoint of the XGS-sequence the terms at z-orders of 694, 706, and 802 are
wrongly reported because these are not yet "fully filled".
XGS:=
.......1........4.........9.......16......67.....36...115...64....365...100...405...144...573...196..1407...256
O(----) + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---- + ---
.....498........4..........6........8......10.....12.....14.....16.....18.....20....22....24.....26.....28......30.....32
.....z..........z............z........z.......z.......z........z.......z.......z........z......z.......z.......z.........z........z.......z
693611...29584..391073..30276..513331..30976..791897..31684..375211..32400..299815
+ ------ + ----- .+ ------. + ----- + ------ .+ ----- + ------ .+ ----- .+ ------ + ----- + ------
......342.......344........346......348.......350.......352.......354.......356........358......360......362
.....z...........z.............z...........z...........z............z...........z............z.............z..........z..........z
33124..782949..33856..624309..34596..488397..35344..1058773..36100..424811..36864
+ ----- + ------ + ----- + ------ + ----- + ------ ..+ ----- + ------- .+ ----- + ------ .+ -----
....364........366......368........370......372........374......376........378........380........382......384
...z............z..........z............z...........z............z..........z..............z.............z...........z..........z
3855760..2315912..2331240..5972526..2491588..1938218..4864462..2579940..2630384
+ ------- + ------- + ------- ..+ ------- .+ ------- .+ -------..+ ------- ..+ ------- ..+ -------
.......678..........682..........686..........690..........694..........698..........702..........706.........710
......z..............z..............z..............z..............z...............z...............z...............z..............z
5802612..4197900..3080336..8089508..3045198..3388736..8412822..3926768..3273776
+ ------- + ------- ..+ -------. + -------. + ------- .+ ------- .+ ------- .+ ------- .+ -------
.......786..........790..........794..........798.........802.........806...........810.........814..........818
......z..............z..............z..............z..............z..............z................z..............z..............z
.................................................................................................(9).
Equations (7) to (9) are only suitable for generating the XGS-sequence. If it is
desired to perform primality tests on individual numerator coefficient, then the procedures
shown below have to be used:
Test (1): First XGS-prime = 67/z^10. Therefore i = 5.
x1:=sort(2*sum(sum((i+2*j)*(i-2*j)/z^(2*i)*(isprime(i+2*j)-false)*(isprime(i-2*j)-false)/(true-false)^2,
i=5..5),j=1..5/2));
....................42
..........x1 := ---
.......................10
.....................z
x2:=sort(sum(i*i/z^(i+i),i=5..5));
....................25
..........x2 := ---
.......................10
....................z
XGS:=x1+x2;
......................67
..........XGS:= ---
..........................10
........................z
Put z:= 1; Then isprime(67) ---> True.
Test (2): 2611997/z^694, i = 694/2 = 347.
..................2491588
..........x1 := -------
..........................694
.......................z
.................120409
..........x2 := ------
........................694
.......................z
...................2611997
..........XGS := -------
............................694
.........................z
Put z:= 1; Then isprime(2611997) ----> True.
4. Summary
This paper defines a new number sequence called the Extended Goldbach's Sequence
(XGS) and develops a sequence algebraic function to generate the numbers. The reason
for calling it XGS is because it is based on the extension to the Goldbach's sequence.
XGS has no connection with the Extended Goldbach Conjecture (XGC) hosted by another
URL-site. XGS primes are quite rare but not as rare as Mersenne's primes. There
is a geometrical basis for XGS primes which makes it easier to predict than Mersenne's
primes.
6. Reference:
[Comments: Papers included in this section are relevant for background readings
but are not directly referenced in the main test. 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. The Canonical Generating Function
or CGF(z) ... - by Huen Y.K.
(date released : 27.5..97) (24 KBytes, 7*A4s).
========================================================
5. 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).
========================================================
6. Lemmata, Corollaries, And
Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).
========================================================
7. Some Interesting Contiguity
Properties Of Odd(z)^2 - by Huen Y.K. (date released : 15.8.97) (36.3 KBytes, 10*A4s).
========================================================
8. Improved Formulations For Comp(z)
and Prime(z)
- by Huen Y.K. (date released : 16.9.97) (17 KBytes ).
========================================================
9. 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 ).
========================================================
10. The Throwing Power Of
Oddcomp(z).
- by Huen Y.K. (date released : 24.9.97 ) (15 Kbytes).
========================================================
11. Sequence Algebraic
Approach To Prime Number Theorem
- by Huen Y.K. (date released : 28.9.97 ) (21 Kbytes).
========================================================
12. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
========================================================
13. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
========================================================
14. In Search Of Counter-
Examples In Maple's Isprime Function.
- by Huen Y.K. (date released : 4.10.97 ) (18 Kbytes).
========================================================
15. A Sequence Algebraist's
View Of Lehmann's Primality Test
- by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).
========================================================
16. On Odd(z), Oddcomp(z),
Seq1(z) and Seq2(z)
- by Huen Y.K. (date released : 10.10.97 ) (17 Kbytes).
========================================================
17. How To Generate A Short
And Contiguous Oddcomp(z) Sequence?
- by Huen Y.K. (date released : 15.10.97 ) (13 Kbytes).
======================== END OF PAPER =========