Evaluations Of Normc( ) Function In Macsyma 2.2
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: 17/12/97)
Abstract
One of the functions unique to sequence algebra in number theory is Normc( ). This function
is defined to reduce all numerator coefficients to unity values. In sequence algebra, any
number sequence with unity numerator coefficients is called a normalised sequence.
Primitive sequences such as Nat(z), Even(z) and Odd(z) are normalised sequences. On the
other hand, Comp(z), Oddcomp(z) and Prime(z) are not normalised but can be normalised
by applying the Normc( ) function. Macsyma 2.2 can be programmed to implement the
Normc( ) function which greatly increases the manipulative power of sequence algebraic
functions using global sequences as variables. The remaining operators still awaiting
implementation are concerned with inter-sequence logical operations but some operations
can now be replaced by using the Normc( ) function. This is desirable since sequences
containing boolean logic operators cannot be handled as global variables within generating
functions.
1. Introduction
A necessary condition recognised quite early in the development of sequence algebra is that
the manipulations of global sequences as variables within generating functions cannot be
handled seamlessly unless these sequences are in normalised forms. Previously this
presented a problem since primitive sequences such as Nat(z), Even(z), and Odd(z) are
normalised but higher ordered sequences derived from the formers such as Comp(z),
Oddcomp(z) and Prime(z) are not. The symbolic software Macsyma 2.2 with its rich set of
mapping functions makes it possible to implement the Normc( ) function and thus the barrier
between the two groups of sequences has finally been eliminated. This paper reports briefly
how the Normc( ) function is programmed and how it can be used to derive the closed
sequence algebraic formulations for Oddcomp(z), Prime(z) and Goldbach(z). In this paper,
we continue to use notations originally adopted in Maple V R 3 but a separate set of symbols
is reserved for Macsyma. Here are some examples how the substitutions are made:
Oddcomp(z) in Maple is made equivalent to Oddcomp_z in Macsyma and similar Prime(z)
in Maple to Prime_z in Macsyma. Both types of symbols are used interchangeably in this
paper but confusion is unlikely to arise.
2. Generating Function For Prime(z)
Equations (2) to (4) show the expansions of three normalised number sequences which are
also called primitive sequences. These are programmed and computed using Macsyma 2.2
with the equivalent Maple V R 3 lines included within square brackets beneath them. Since
the Normc( ) function has yet to be implemented in Maple V R 3, the Normc( ) function in the
Maple line is conceptual only].
Nat_z:taylor(z/(z-1),z,inf,20);
[Maple: Nat(z):=series(z/(z-1),z=infinity,20);]
.................1........1........1.........1........1........1........1........1........1.......1......1.......1......1......1
1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + ---
...................2........3........4.........5........6.........7.......8........9.......10.....11.....12.....13....14....15
.................z........z.........z.........z.........z.........z........z.........z........z........z.......z.......z......z.......z
.
...1......1.......1.......1......1
+ --- + --- + --- + --- + --- ............................................................(1).
......16.....17....18.....19.....20
.....z.......z.......z.......z.......z
Even_z:taylor(z^2/(z^2-1),z,inf,20);
[Maple: Even(z):=series(z^2/(z^2-1),z=infinity,20);]
........1.........1........1........1.......1.......1......1.......1.......1.......1
1 + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- ............(2).
..........2..........4........6........8......10.....12.....14.....16.....18.....20
........z..........z........z........z........z........z.......z.......z.......z........z
Odd_z:taylor(z/(z^2-1),z,inf,20);
[Maple: Odd(z):=series(z/(z^2-1),z=infinity,20);]
...........1.......1.........1........1........1......1.......1.......1.......1
1/z + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- ..................(3).
.............3........5.........7........9.......11....13.....15.....17.....19
...........z........z.........z........z.........z......z.........z.......z.......z
"series" here represents the Comp(z)-sequence computed using the Equation of Composites
as shown in equation (4)[2,3].
series:taylor(sum(1/(z^i*(z^i-1)),i,2,20),z,inf,20);
[Maple: Comp(z):=series(sum(1/(z^i*(z^i-1)),i=2..20),z=infinity,20);]
..1........2.........2........1.......2.......4......2......2.......3.......4........4
---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- .................(4).
....4........6.........8........9......10.....12.....14.....15.....16.....18.....20
..z.........z.........z........z.......z........z.......z.......z........z.......z........z
Normc( ) function is defined in equation (5):
Normc(series):=block([args:args(series)],apply("+",args/map('numfactor,args)))$ ...(5).
[Maple: Normc( ) is only conceptual for the time being awaiting implementation.]
In equation (6), the Comp(z)-sequence from equation (4) is normalised.
Comp_z:Normc(series);
[Maple: Same remark as in the previous line.]
..1........1.........1........1.......1.......1.......1......1......1.......1.......1
---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- ......................(6).
....4........6.........8........9.......10....12.....14.....15....16....18....20
..z.........z.........z........z........z.......z.......z........z.......z.......z.......z
The normalised Prime(z) sequence shown in equation (7) cannot be obtained without a
normalised Comp(z)-sequence from equation (6).
Prim_z:Nat_z-Comp_z -1 -1/z;
[Maple: Prime(z):=Nat(z)-Comp(z)-1-1/z;]
This normalised Prime(z)-sequence can be manipulated further within generating functions
exemplified by the derivations of Oddcomp(z) and Goldbach(z) which follow.
..1........1........1........1........1.......1.......1.......1
---- + ---- + ---- + ---- + --- + --- + --- + ---......................................(7).
....2........3........5........7.......11.....13.....17......19
..z........z........z.........z........z........z.......z.........z
Once we get the normalised Prime(z), we can derive the normalised Oddcomp(z) sequence
as follows:
Oddcomp_z:Odd_z-Prime_z-1/z -+1/z^2;
[Maple: Oddcomp(z):=Odd(z)-Prime(z) - 1/z + 1/z^2;]
................................................1........1
..............................................---- + --- ..................................................(8).
..................................................9.......15
................................................z........z
3. Goldbach's Sequence And Goldbach's Conjecture
Goldbach's Conjecture states that every even positive integer greater than 3 is the sum of
two (not necessarily distinct) primes. The definiton of Goldbach's sequence given by
equation (9) gives us the opportunity to study Goldbach's Conjecture algebraically. In other
words if we can prove that Goldbach(z) is globally contiguous, then Goldbach's Conjecture is
proved. The last contiguous Goldbach(z) was found by the author when the prime-set
{5,7,.......,109} was used as Prime(z). Vincent Celier of Cananda has scanned numerically
for contiguous prime-sets all the way up to 1,000,000,000 without discovering any contiguous
sets [ ref: section 1 of this Website.]. The above ceiling looks rather low but remember
that each test was down with a full contiguous
prime-set and as this set increases in lengths, the computations can be very arduous. Lately,
the author has proved algebraically that the contiguity of Odd(z)^2 will always be broken by
just removing the second largest odd number from the Odd(z)-sequence [11]. Since Prime(z)
is a subset of Odd(z), it is the nearest to a proof that Goldbach's Conjecture is false. But
there may be a flaw in it -- how does one handle infinity? Most probably, either Goldbach's
Conjecture is false or unprovable.
Goldbach(z) has been previously defined as shown in equation (9):
Goldbach(z) := Normc(Prime(z)^2); ........................................(9).
Equation (10) gives the normalised expansion of Goldbach(z):
Goldbach_z:Normc((Prim_z-1-1/z-1/z^2-1/z^3)^2);
..1......1.......1......1.......1........1........1......1
--- + --- + --- + --- + --- ..+ --- + --- + --- .........................(10).
..10.....12.....14....16.....18......20.....22.....24
.z........z.......z.......z.......z........z........z.......z
Continuation of the search for Goldbach's sequence larger than that found by Vincent Celier
is hampered by finite precision of compiled languages. One suggestion by the author is to
compute backward using a symbolic package since loss of contiguities always occurs at the upper extreme ends of
expansions. This is a project the author intends to undertake using Macsyma 2.2 shortly.
4. Summary
This paper reports briefly how Macsyma 2.2 can be used to program the Normc( ) function
which will greatly increase the manipulative power of generating functions in sequence
algebra. Since all symbolic langauges are rich in mapping functions, there is no doubt that this
function can be implemented by all of them. Lack of familiarities with programming syntices
is the main source of delay in implementations.
5. 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).
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