1. Introduction

This problem was originally posted as a challenge in this website on 1.9.97 and closed on 31.10.97. No correct solutions have been received. An algebraic proof based on order analysis is presented by the author in this short paper. This is proved as a theorem.

Theorem 1: If the second largest odd or second smallest odd term from Odd(z) is removed before squaring this sequence, the resultant even sequence will never be contiguous.

Loss of contiguity needs only be the loss of one term in the original sequence. This proof is made possible because we confine to the extremities of the sequence. In this proof, we only consider the upper end. The same method can be used for the lower end. For convenience, we take arbitrarily a sample of five consecutive odd terms in the upper end of Odd(z).

Proof: Take a general expression of the last five terms in Odd(z) as shown in equation (1).

.............................a(2 k - 7)....a(2 k - 5)....a(2 k - 3).....a(2 k - 1).....a(2 k + 1)
..............Odd(z) := ---------- + ---------- + ---------- + ---------- + ---------- ...............(1).
................................(2 k - 7).....(2 k - 5).....(2 k - 3)......(2 k - 1)......(2 k + 1)
...............................z................z.................z.................z..................z

Then squaring Odd(z) we get equation (2):

...................14..............2......12....................................10
..................z...a(2 k - 7).......z...a(2 k - 7) a(2 k - 5).....z....a(2 k - 7).a(2 k - 3)
Oddsq(z) := ----------- + 2 -----------------------+ 2 -------------------------
...........................k 4............................k 4................................k 4
......................(z )............................(z )................................(z )

..............10..............2......8.....................................8
............z....a(2 k - 5)......z..a(2 k - 7) a(2 k - 1).....z..a(2 k - 5).a(2 k - 3)
........+ --------------+ 2 ---------------------- + 2 ------------------------
......................k 4..........................k 4..............................k 4
.................(z )..........................(z )..............................(z )

................6.....................................6..................................6..............2
..............z a(2 k - 7) a(2 k + 1)......z a(2 k - 5) a(2 k - 1)...z a(2 k - 3)
........+ 2 ---------------------- + 2 --------------------- + --------------
.............................k 4.................................k 4..........................k 4
........................(z ).................................(z )..........................(z )

................4.....................................4...................................2
..............z a(2 k - 5) a(2 k + 1)......z..a(2 k - 3) a(2 k - 1)....z..a(2 k - 3) a(2 k + 1)
........+ 2 ----------------------+ 2 ----------------------+ 2 ------------------------
.............................k 4..................................k 4.................................k 4
........................(z )..................................(z ).................................(z )

..............2..............2.....................................................2
............z a(2 k - 1)......a(2 k - 1) a(2 k + 1)...a(2 k + 1)
..........+ ----------- + 2 --------------------+ ----------- .........................................(2).
.......................k 4.......................k 4................2.....k 4
..................(z ).......................(z )...................z..(z )

It will be noted from the second largest term (in bold fonts) in equation (2) that the numerator expression is given by the product of a(2*k-1) and a(2*k+1). In other words, the second largest term depends only on the product of the numerators associated with the last and second last term of Odd(z). Therefore if array a(2*k-1) is equated to zero, this term will be reduced to zero thus breaking contiguity. There cannot be further breaks in contiguities further down the sequence since this is Odd(z) but for Prime(z) there could be additional breaks. This is an algebraic proof since no specific values are assigned to k. Furthermore, it is impossible for lower terms other than the second last and last term to contribute to the numerator of the second last term. Q.E.D.


3. Conclusions

Sequence and order are important number theoretic properties which do not seem to receive much attention from conventional number theory. Theorem 1 could be used to explain qualitatively why there are so few full Goldbach sequences. In fact, beyond GS1(109):0 no further full GS has been found up to the range of 1000,000,000. Although this was not a very large search range, the present theorem hints at the rarity of full GS's in the large integer end. The increasing gaps between primes do not favour the formatons of full GS's.


4. References

(Comments: Not all papers are directly relevent but are included for the benefit of readers who are new to sequence algebra.)

1. A Simple Introduction To Sequence Algebra - by Huen Y.K. (date release: 15.3.97) (38 KBytes, 11*A4 pages).

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2. The Canonical Generating Function or CGF(z) ... - by Huen Y.K. (date released : 27.5..97) (24 KBytes, 7*A4s).

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3. Final Value Theorem Applied To Number Sequences... - by Huen Y.K. (date released : 5.6.97) (29.4 KBytes, 9*A4s).

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4. Unsolved Problems In Sequence Algebra - by Huen Y.K. (date released : 6.6.97) (29.4 KBytes, 9*A4s).

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5. Lemmata, Corollaries, And Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).

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(6) The following articles are either published or in the press but for copyright reasons, these cannot be reproduced here for download. Please contact the publishers: Taylor & Francis Ltd., 1 Gunpowder Square, London, EC4A 3DE, England, for further information:

(i) Some interesting properties of the natural number system, INT.J.MATH.EDUC.SCI.TECHNOL, 1996,VOL.27,NO.5,685-691.

(ii) Hgram patterns of Routh stability zones in linear systems, INT.J.MATH.EDUC.SCI.TECHNOL, 1997,VOL.28,NO.2,225-241.

(iii) Visual algebra and its applications, INT.J.MATH.EDUC.SCI.TECHNOL, 1997,VOL.28,NO.3, 333-344.

(iv) The twin prime problem revisited, INT.J.MATH.EDUC.SCI.TECHNOL.,199?,VOL.??, NO.?,???-???, proof paper mes-0488 (10 pages).

(v) Is Pie Periodic?, INT.J.MATH.EDUC.SCI.TECHNOL.,199?,VOL.??,NO.?,???-???, (in the press).

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7. Some Interesting Contiguity Properties Of Odd(z)^2 - by Huen Y.K. (date released : 15.8.97) (36.3 KBytes, 10*A4s).

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8. What Is So Special About GS1(109):0 - by Huen Y.K. (date released : 16.8.97) (29.3 KBytes, 9*A4s).

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