Goldbach's Sequence And Goldbach's Conjecture
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: 18/12/97)
Abstract
Goldbach(z) represents Goldbach's sequence and is defined by the author as: Goldbach(z):=
Normc(Prime(z)^2). Finite lengths of Goldbach(z) do not prove or disprove Goldbach's
Conjecture but if the global Goldbach's sequence is contiguous, then
Goldbach's Conjecture will be true. The longest contiguous Goldbach(z) discovered so far
is GS1(109):0 which means that the square of the prime-set {5,7,...,109}
will form a contiguous even integer set given by {10,12,14,.......,216,218}. No further
contiguous Goldbach(z) has been found from systematic searches using prime-sets within the integer
range just below 1,000,000,000 by Vincent Celier of Canada. Computatons can get very arduous
with increasing lengths of prime-sets and are hampered by limited precisions of compiled
programs. A determinstic shortcut for testing noncontiguity of Goldbach(z) is described in this
paper. To determine whether a specific Goldbach(z) is contiguous, we still need to perform
global tests.
1. Introduction
A very efficient way of weeding out unnecessary tests for noncontiguities in Goldbach's
sequences, i.e. Goldbach(z), is to test only the high ends of Prime(z). This comes from a
theorem on the contiguity of Odd(z)^2 in which it was proved that if the second largest odd
integer is removed from Odd(z) before squaring, the resultant even integer sequence is
never contiguous [11]. Since Prime(z) is a subset of Odd(z), we know that if Odd(z)^2 is not
conitiguous then Prime(z)^2 of the same integer range will not be contiguous. This method
is used here to extend the range of search for noncontiguous Goldbach(z) above 10^9. The
method is determinstic on noncontiguities only. To determine contiguities, we still need to
perform the full contiguity tests.
2. The Original Global Contiguity Tests
We will recall briefly the original contiguity test on Goldbach(z). Since we already found that
GS1(109):0 is contiguous, we will apply global tests on Prime(z) close to this range. All
computations will be performed using Macsyma 2.2 supplemented by equivalent program
lines using Maple V R 3 for the benefit of Maple symbolic software users.
(i)Contiguity Test On GS1(109):0
Prime_z:sum(1/z^i*(primep(i)-false)/(true-false),i,5,109)
[Maple: Prime(z):=sum(1/z^i*(isprime(i)-false)/(true-false),i=5..109);]
..1.......1.......1......1.....1......1......1.......1.......1.......1......1.......1.......1.......1.......1......1......1
----+ ----+ ---+ ---+ ---+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
....5.......7.....11....13...17.....19....23.....29.....31.....37.....41....43.....47.....53.....59.....61....67
.z........z.......z......z......z.......z.......z.......z.......z........z.......z.......z.........z.......z.......z......z.......z
.....1......1.......1.......1.......1.......1.......1.......1.........1........1
+ --- + --- + --- + --- + --- + --- + ---- + ---- + ---- + ---- ...........................(1).
......71....73.....79.....83.....89.....97....101....103.....107....109
.....z......z.......z........z........z.......z.......z........z..........z.........z
Goldbach_z:expand(%*%)
[Maple: Goldbach(z):=sort(expand(Prime(z)^2));]
.1......2.......1.......2.......4......2......3.......6.......3.......4.......6.......2......5......8......3.......4.........8
--- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...10....12.....14.....16....18.....20....22.....24.....26....28.....30.....32....34.....36....38.....40......42
.z.......z.......z.......z........z.......z......z........z.......z.......z.......z........z.......z......z.......z........z........z
....4.......5....10......6........6......10.....4......7......12......3.......8......12......4.......8.....12......7
+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.....44.....46.....48.....50.....52.....54....56.....58.....60.....62.....64.....66....68.....70....72.....74
....z.......z.......z......z.........z........z......z.......z.......z........z.......z........z........z.......z.......z.......z
....8......14.....8......7.......16......7......8......18......6......9......14......6......10......16.......8
+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---- + ---- + ----
.....76.....78.....80.....82.....84.....86....88.....90.....92.....94.....96.....98....100....102....104
....z.......z.......z.......z........z........z......z........z.......z.......z.......z........z........z........z........z
....9......16.......10.......12.......20.....10.......9.......22.......7........8........18.........6.........10
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----
.....106....108....110....112....114....116.....118....120....122.....124.....126....128....130
....z.......z.........z.........z.........z........z..........z.........z.........z.........z........z.........z.........z
....14......7.......6........12.......10.......7.......12.......7........6........14........4........6........10
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----
......132...134....136....138....140....142.....144....146....148......150....152....154....156
.....z.......z........z.........z.........z.........z.........z.........z.........z.........z........z.........z.........z
.....3.......6.......10.......4........3........10.......6........4........6........6.........3........8.........4
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----
......158....160....162....164....166....168.....170....172....174.....176....178....180....182
.....z........z........z........z.........z.........z.........z.........z........z.........z.........z.........z.........z
.....2.......6.......2.........4........4........1........2.........4.........2........1........4.........3........2
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----
.....184....186....188....190....192.....194....196....198.....200....202....204......206.....208
.....z.......z........z.........z.........z.........z.........z.........z.........z.........z.........z.........z........z
....4........2.........1.........2.........1
+ ---- + ---- + ---- + ---- + ---- ..........................................................(2).
.....210....212.....214.....216.....218
....z........z.........z..........z..........z
A visual check of the power indices of the z-order variables will show that this is a contiguous
even integer set. This sequence is unnormalised but this can be normalised using the
Normc( ) function previously developed [12].
(ii)Contiguity Test On GS1(107)
Prime_z:sum(1/z^i*(primep(i)-false)/(true-false),i,5,107)
[Maple: Prime(z):=sum(1/z^i*(isprime(i)-false)/(true-false),i=5..107);]
For page economy, the resultant Goldbach(z) sequence is shown partially in equation (4).
Only the initial stretch and the last stretch are shown and there are two even integer gaps at 1/z^188 and
1/z^212. This Goldbach(z) is noncontigious.
..1......2......1.......2.......4.......2.......3.......6.......3.......4.......6......2......5.......8......3.......4......8
--- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...10....12....14.....16.....18.....20.....22.....24.....26.....28.....30.....32....34.....36.....38.....40....42
..z......z.......z........z........z.......z......z........z.......z.......z........z........z.......z.......z.......z.......z.......z
+.............................................................+...................
.....2........6.......4........2........1.........2.........2.........2........1.........4........1........2........2
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----
......184....186....190....192....194....196.....198.....200....202....204....206....208....210
.....z........z.........z........z.........z.........z.........z.........z........z.........z........z.........z........z
......1
+ -------- ......................................................................................(4).
........214
......z
(iii)Contiguity Test On GS1(113)
Prime_z:sum(1/z^i*(primep(i)-false)/(true-false),i,5,113)
[Maple: Prime(z):=sum(1/z^i*(isprime(i)-false)/(true-false),i=5..113);]
For page economy, the resultant Goldbach(z) sequence is shown partially in equation (5).
Only the initial stretch and the last stretch are shown and there is only one gap at 1/z^224 in
this Goldbach(z) which is noncontiguous.
..1.....2.......1......2.......4.......2......3........6......3.......4.......6......2.......5......8.......3.......4......8
--- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...10....12.....14....16....18......20.....22.....24.....26....28.....30.....32....34....36.....38.....40....42
..z.....z........z......z.......z........z.......z........z.......z........z.......z.......z.......z......z........z........z......z
+............................................................+................
....6........2........3........4........1........2.........2.........1
+ ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- ..........................................(5).
......210....212....214....216....218....220....222....226
....z........z.........z........z.........z.........z.........z.........z
3. The Short-Cut Method
In this method, contiguity of Goldbach(z) is only tested using the high ends of the Prime(z)-
sequence. The method is based on the following reasonings:
(i) Loss of contiguities is found to concentrate in the second half of Goldbach(z) and in actually fact
close to the upper end of the sequence. Once noncontiguity is detected at the
upperend, the result is deterministic.
(ii) Given below are the last four terms in the Odd(z) sequence:
.....1..............1..................1................1
---------- + ---------- + ---------- + ---------- .............................(5).
....(2 n - 5).....(2 n - 3)....(2 n - 1)....(2 n + 1)
..z...............z................z................z
Taking the square of the above sequence, we get:
......1.............2...............3...............4...............3.................2...............1
--------- + --------- + -------- + --------- + --------- + --------- + -------- ........(6).
.....4n-10........4n-8..........4n-6...........4n-4..........4n-2............4n...........4n+2
(z ).............(z )............(z ).............(z )............(z )..............(z )..........(z )
It can be seen from equation (6) that if the second largest term in equation (5) is removed
before squaring, the second largest term in the former equation will be eliminated thus
creating noncontiguity in Goldbach(z). We only need to use at most the last four prime terms
at the upper end of Prime(z). The closest these four primes can pack will consist of
one isolated prime and a pair of twin-primes. It is impossible to have two pairs of twin-primes
in close succession. If Prime(z) is
capped by a twinprime pair, one will have to look for gaps further down along Goldbach(z). If
Prime(z) is capped by an isolated prime, it will certainly be noncontiguous. In this shortcut
test, for safety we use the last 20 primes in Prime(z).
(iii) If the stretch of Goldbach(z) is contiguous, then it is recommended that a global
contiguity test be performed on this specific sequence.
4. Outline Of Search Beyond In The High Range
Since search below 10^9 has already been completed, we give here an example of testing in
the range of 10^20 using a program line written in Macsyma 2.2 shown in equation (7). By
applying a primality filter over 500 integers, we get the 20 primes at the high end. The
use of probabilistic primality tests to set up Prime(z) is justifiable since such tests never
report a prime as a composite but may report a composite as a prime. Even if a false
prime is admitted into Prime(z), it will not be able to switch noncontiguity to contiguity
in Goldbach(z) especially at the large end of the number system. The probability of
getting more than one false report in 20 primes is almost nil.
Prime_z:sum(1/z^(2*i+1)*(primep(2*i+1)-false)/(true-false),i,10^20,10^20+500)
[Maple: Prime(z):=sum(1/z^(2*i+1)-false)/(true-false),i=10^20..10^20+500)
d:\Macsyma\Macsyma2\library1\binoml.fas being loaded.
Prime(z):=
{200000000000000000089, 200000000000000000137, 200000000000000000209,
200000000000000000233, 200000000000000000261, 200000000000000000309
200000000000000000329, 200000000000000000341, 200000000000000000347
200000000000000000459, 200000000000000000467, 200000000000000000597
200000000000000000629, 200000000000000000641, 200000000000000000677
200000000000000000789, 200000000000000000831, 200000000000000000837
200000000000000000839, 200000000000000000993}.....................................(7).
Taking the square of the lopped Prime(z) sequence in equation (7) we get the lopped
Goldbach(z) as shown in equation (8). Examining the largest two terms, we already know that
this is noncontiguous and need to look no further. Note that gaps between primes
increase with the integer range used so that probabilities of achieving contiguous
Goldbach(z) fall with increasing range of search.
........................1..........................................2..........................................1
------------------------------ + --------------------------- + ----------------------------
...400000000000000000178....400000000000000000226....400000000000000000274
z..............................................z...........................................z
+...................................................................+...........
........................2..........................................2..........................................2
+ ----------------------------- + ---------------------------- + ----------------------------
...400000000000000001782......400000000000000001824....400000000000000001830
z................................................z............................................z
........................2..........................................1
+ ----------------------------- + --------------------------- .....................................(8).
...400000000000000001832.......400000000000000001986
z.................................................z
5. Summary
This paper outlines a shortcut by which multidigit arithmetic in symbolic softwares
can be exploited to test noncongituities of Goldbach(z) over very large integer range.
It is suggested that no more than 20 primes in the upperend of Prime(z) need be
taken. The test is deterministic on noncontiguities of Goldbach(z). If the test shows
that this upper stretch of even integers is contiguous, then a full contiguity test will
have to be taken. Being a numerical test, it cannot be used to prove directly Goldbach's
Conjecture. However findings on Goldbach(z) might provide the backdrop for proving
Goldbach's Conjecture by an as yet undiscovered method. One cannot rule out that
perhaps Goldbach's Conjecture is unprovable.
6. Reference:
[Comments: Not all papers are directly referenced in the main text.
Some are included as these are relevant for background readings. You can hyperlink to most of these
papers from within this reference section. ]
========================================================
1. Macsyma Mathematics and System Reference Manual (Sixteenth Edition) 1996. Section
15.3, pp 404-405.
========================================================
2. Huen Y.K.:
A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994,
Vol.25, No.6, pp 913 - 920.
=======================================================
3. Huen Y.K.:
Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997,
Vol.28, No.3, 333-344.
=======================================================
4. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
========================================================
5. The Canonical Generating Function
or CGF(z) ... - by Huen Y.K.
(date released : 27.5..97) (24 KBytes, 7*A4s).
========================================================
6. Information Contents Of Number
Theoretic Functions - by Huen Y.K. (date released : 29.5.97) (21.5 KBytes, 7*A4s).
========================================================
7. Visual Solutions Of Number Theoretic
Problems ..... - by Huen Y.K. (date released : 3.6.97) (38.3 KBytes, 10*A4s).
========================================================
8. Information Contents Of
Hypothetical DNA Sequences - by Huen Y.K. (date released : 27.6.97) (26.0 KBytes, 8*A4s).
========================================================
9. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
========================================================
10. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
==============================================
11. Algebraic Proof To A
Problem Posted In Challenge 2
- by Huen Y.K. (date released : 3.11.97, revised 4/11 ) (13 Kbytes).
=====================================================
12. Evaluations Of Normc( ) Function In Macsyma 2.2
- by Huen Y.K. (date released : 17.11.97 ) 14 Kbytes).
======================== END OF PAPER ====================