1. Introduction
Somehow connected with symmetry breaking after the Big-Bang, we have to contend with the
presence of asymmetry in the natural number system. In sequence algebra, the presence or
absence of an integer is indicated by the numerator coefficient associated with each expanded
term. This coefficient is also called the duplicity factor being a counter of the number of copies
of each integer. Although not obvious, we can view a number sequence as a divisibility set. If N
represents the global number set then Nat(z) arises from the divisibility set of {N/1} and Even(z)
from {N/2} if we assume that 0/1 and 0/2 are divisibles since these leave no remainders. A direct
divisibility test for Odd(z) does not exist unless we resort to detecting the residue set of 1s and 0s in N mod
2. Although modular arithmetic is powerful it is not an operator one would like to see mixed up
in ordinary arithmetic expressions since it is a one-way function [1]. A truly consistent arithmetic
domain should admit the use of such operators as Abs, Sgn, Max, Min, Mod and Normc, and
even the boolean logic set but this ideal situation is still a pipedream [5]. Therefore Odd(z) is
derived indirectly from two known divisibility sets shown as follows:
......................Odd(z) = Nat(z) - Even(z) .................................................(1)
or alternatively we can derive Odd(z) by performing an interval shift of Even(z) by one unit
interval to the right as shown in equation (2).
......................Odd(z) = Even(z)/z ............................................................(2).
One must be specific on whether the number sequences should include the integer 0 and if
it does, this is done by defining Nat(z) and Even(z) as follows.
........................Nat(z) = z/(z-1) ...................................................................(3),
and...................Even(z) = z^2/(z^2-1) ........................................................(4).
Thus we have to recognise that it is much easier to detect divisibility in sequence algebra than
indivisibility. There is asymmetry in these two properties. This is the reason why the direct formulation of Prime(z) without relying on
Comp(z) is an unsolved problem in sequence algebra [8].
2. Periodicities In Number Sequences
Curve-fitting in the discrete domain is almost parallel to Fourier's harmonic analysis in the
continuous domain. The major difference is that whereas pure sinusoidal waveforms are used in
the latter, periodic impulse sequences are used in the former. Because an impulse sequence
can be modelled by an infinite number of sinusoidal waveforms, it is a far richer curve-fitting
component than the latters.
This paper explains various techniques in curve-fitting the composite and the prime sequence. A
general primitive term in a sequence is defined to take the form as shown in equation (5).
.......................................................................k
......................................Generalterm(z) := ----- ................................(5).
..........................................................................i
.......................................................................z
k is a numerator coefficient which is called the duplicity factor, i.e., the counter for the number of
duplicates of an integer i. The exponent i to the order variable z which appears in the
denominator represents the integer i itself which is always taken as a member of a holistic
number sequence which ranges from zero to infinity. In fact i is a measure of the number of unit
intervals displaced from the first integer in the global set. One must recognise that the 1s in a
periodic sequence represent divisibilty whilst the 0s represent indivisibility. Thus the foundation
of sequence algebra is based on divisiblity sequences expressed as periodic sequences.
The equation of divisibles or composites called Comp(z) shown in equation (6) will be used as an
example to demonstrate the curve-fitting method. We already have a closed generating
function for Comp(z) which was derived by the author using a geometric model [11]. Here we pretend
that we do not know the existence of this closed formulation and make attempts to derive it
manually without using the original matrix model given by equation (6).
.................................................................1
..................................Comp(z) := ------------------ ..............................................(6).
...............................................................i.......i
.............................................................z.....(z - 1)
Equation (6) can be expanded in Laurent's series using the program line written in Maple V R 3
where the output sequence is shown in equation (7). Note that this is an unnormalised series
with values of most numerator coefficients greater than unity. Each numerator coefficient
represents the number of divisors in the composite number i. For example 4 = 2*2 and
25=5*5 have one divisor each whereas 6=2*3 has two divisors.
Comp(z):=series(sum(1/(z^i*(z^i-1)),i=2..60),z=infinity,60);
........................1......2.......2.......1.......2.......4.......2.......2.......3.......4.......4.......2.......2.......6
Comp(z) := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.........................4.......6.......8.......9......10.....12.....14.....15.....16.....18.....20.....21.....22.....24
........................z.......z........z........z......z........z.......z........z.......z........z.......z........z........z.......z
..............1.......2......2.......4......6......4......2.......2.......2.......7......2......2.......6.......6.......4.......4
........+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
..............25.....26....27....28......30.....32....33.....34.....35......36.....38.....39.....40.....42...44....45
..............z......z.......z.......z........z.......z.......z.......z.......z........z.........z.......z.......z.......z.......z......z
...........2........8......1......4........2......4......6.......2.......6.......2......2.............1
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + O(-------) .................(7).
............46......48....49.....50......51....52....54....55......56.....57....58...........60
...........z........z.......z.......z........z.......z......z........z........z......z.......z.............z
Before attempting curve-fitting, one should always test whether the output sequence is
factorisable. Much labour is saved if it is factorisable into simpler polynomial expressions.
The unnormalised Comp(z) sequence in equation (8) is not factorisable as shown in equation
(10).
sort(factor(Comp(z)));
......54......52.....50.....49......48......46.......44.......43.......42.......40.......38.......37.......36
..(z ..+ 2 z ..+ 2 z ..+ z ..+ 2 z ..+ 4 z ..+ 2 z ..+ 2 z ..+ 3 z ..+ 4 z ..+ 4 z ..+ 2 z ..+ 2 z
.............34.....33......31.......30.......28......26........25.......24.......23.......22........20
......+ 6 z ..+ z ..+ 2 z ..+ 2 z ..+ 4 z ..+ 6 z ..+ 4 z ..+ 2 z ..+ 2 z ..+ 2 z ..+ 7 z ..+ 2 z
..............19.......18.......16.......14.......13.......12......10.....9........8........7..........6........4........3
.......+ 2 z ..+ 6 z ..+ 6 z ..+ 4 z ..+ 4 z ..+ 2 z ..+ 8 z ..+ z ..+ 4 z ..+ 2 z ..+ 4 z ..+ 6 z ..+ 2 z
..............2...................../ 58
.......+ 6 z ..+ 2 z ..+ 2) / z .......................................................................(8).
.................................../
We also test whether the normalised composite sequence given by Normc(Comp(z)) is
factorisable. Equation (10) shows the normalised Comp(z) sequence is also not factorisable.
Normally, it is seldom possible to factorise a normalised sequence since the product of two
normalised polynomial sequences always generate an unnormalised sequence.
Normc(Comp(z)):=
............1......1......1........1.......1.......1.......1.......1.......1......1.......1.......1.......1.......1.......1
........---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
..............4.......6.......8.........9......10......12.....14....15.....16....18....20.....21....22....24....25
...........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.......1......1.......1.......1.......1
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
............26.....27......28....30....32......33....34.....35....36......38....39.....40.....42.....44.....45.....46
...........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
..........+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- .......................................(9).
................48......49.....50....51.....52.....54....55....56....57.....58
...............z........z.......z.......z.......z.......z........z......z......z......z
sort(factor(Normc(Comp(z)));
...54....52....50....49....48....46...44...43....42....40...38...37....36....34....33...32....31
(z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z
...........30....28....26....25....24....23....22....20....19...18....16...14....13...12....10.....9
.......+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z
...........8.......7......6.....4......3......2............... / 58
......+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ 1) / z .....................................................(10).
.................................................................../
An alternative method of finding the closed form is to curve fit the unnormalised output sequence
with periodic sequences of increasing intervals. The rule is stipulated that all curve fitting will
proceed from left to right and that at any stage of curve fitting, terms to the left of the point must
have already been perfectly curve-fitted. This ensures that curve fitting is not haphazard. As far
as is possible, we do not wish to introduce periodic sequences with negative or mixed signs
unless it is obvious that such a sequence is unavoidable. Table 1 shows the steps in introducing
N/i periodic sequences. N stands for Nat(z) and i stands for the ith divisor. Whenever an integer
is exactly divisible by the divisor i, a 1 is insert at that point, otherwise a 0 is inserted.
.................Table 1 - Manual method of curve-fitting Comp(z) with periodic
.....................................sequences of increasing intervals.
Comp(z):
From eqn(9)..... 00001020212040223040422061224060422270226060442081424062622
N/2:.................. 00001010101010101010101010101010101010101010101010101010101
......................... 00000010111030122030321051123050321260125050341071323052521
N/3:.................. 00000010010010010010010010010010010010010010010010010010010
......................... 00000000101020112020311041113040311250115040331061313042511
N/4:................... 00000000100010001000100010001000100010001000100010001000100
......................... 00000000001010111020211031112040211240114040231051312042411
N/5:.................. 00000000001000010000100001000010000100001000010000100001000
......................... 00000000000010101020111030112030211140113040221051212041411
N/6:................... 00000000000010000010000010000010000010000010000010000010000
......................... 00000000000000101010111020112020211130113030221041212031411
N/7:................... 00000000000000100000010000001000000100000010000001000000100
......................... 00000000000000001010101020111020211030113020221040212031311
N/8:................... 00000000000000001000000010000000100000001000000010000000100
.......................... 00000000000000000010101010111020111030112020221030212031211
N/9:................... 00000000000000000010000000010000000010000000010000000010000
.......................... 00000000000000000000101010101020111020112020211030212021211
N/10: ................ 00000000000000000000100000000010000000001000000000100000000
......................... 00000000000000000000001010101010111020111020211030112021211
N/11:................. 00000000000000000000001000000000010000000000100000000001000
......................... 00000000000000000000000010101010101020111020111030112020211
N/12:................. 00000000000000000000000010000000000010000000000010000000000
......................... 00000000000000000000000000101010101010111020111020112020211
N/13:................. 00000000000000000000000000100000000000010000000000001000000
......................... 00000000000000000000000000001010101010101020111020111020211
N/14:................. 00000000000000000000000000001000000000000010000000000000100
......................... 00000000000000000000000000000010101010101010111020111020111
N/15:................. 00000000000000000000000000000010000000000000010000000000000
......................... 00000000000000000000000000000000101010101010101020111020111
N/16:................. 00000000000000000000000000000000100000000000000010000000000
......................... 00000000000000000000000000000000001010101010101010111020111
N/17:................ 00000000000000000000000000000000001000000000000000010000000
......................... 00000000000000000000000000000000000010101010101010101020111
N/18:................ 00000000000000000000000000000000000010000000000000000010000
......................... 00000000000000000000000000000000000000101010101010101010111
N/19:................. 00000000000000000000000000000000000000100000000000000000010
.......................... 00000000000000000000000000000000000000001010101010101010101.....
After the N/19 stage, a regular pattern of residues consisting of 1010101... has been established
which indicates that most probably subsequent N/i periodic sequences should appear
consecutively without breaks. Since each N/i layer takes the form 1/(z^i*(z^i-1)) we thus recover
the orginal general formulation for Comp(z) given by equation (6). The manual method is
summarised in Table 1. A question which arises is whether one could use the same method on
Prime(z) as shown in equation (11).
....................1......1.......1......1......1.......1.......1.......1.......1.......1.......1.......1.......1......1
Prime(z):= ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
......................5......7.......11....13......17.....19.....23.....29....31....37......41.....43....47....53
....................z......z........z......z.......z........z........z.......z..........z......z........z.......z......z.......z....(11).
..............................Table 2 - Manual Method Of Curve-Fitting Prime(z)
Prime:................ 000001010001010001010001000001010000010001010001000001
1/(z^5-1)........... 000001000010000100001000010000100001000010000100001000
.......................... 000000010011010101011001010001110001010011010101001001
1/(z^5(z^5-1))... 000000000010000100001000010000100001000010000100001000
......................... 000000010001010001010001000001010000010001010001000001
1/(z^7-1)............ 000000010000001000000100000010000001000000100000010000
......................... 000000000001011001010101000011010001010001110001010001
1/(z^7*(z^7-1))... 000000000000001000000100000010000001000000100000010000
......................... 000000000001010001010001000001010000010001010001010001
1/z^11............... 000000000001000000000010000000000100000000001000000000
......................... 000000000000010001010011000001010100010001011001010001
1/(z^11*(z^11-1)) 000000000000000000000010000000000100000000001000000000
......................... 000000000000010001010001000001010000010001010001010001
1/z^13............... 000000000000010000000000001000000000000100000000000010
......................... 000000000000000001010001001001010000010101010001010011
1/(z^13*(z^13-1)) 000000000000000000000000001000000000000100000000000010
......................... 000000000000000001010001000001010000010001010001010001
1/z^17............... 000000000000000001000000000000000010000000000000000100
......................... 000000000000000000010001000001010010010001010001010101
1/(z^17*(z^17-1)) 000000000000000000000000000000000010000000000000000100
......................... 000000000000000000010001000001010000010001010001010001
This example shows that periodic sequences required to do curve fitting are all based on prime
divisors and that after each N/i layer, it is obvious that it is impossible to avoid introducing a
negatively signed N/i layer to remove the extra periodic bits appearing. This is equivalent to
curve-fitting by the canonical generating function CGF(z) where f(i) is expected to supply each
successive prime. This is circular logic in which the method states that "if you want to generate
the prime sequence, supply the prime sequence". It is simply a "beating about the bush" way of
saying that there is no solution. Direct solution of Prime(z) cannot be obtained by curve fitting
the sequence by divisibility sequenes [8].
Currently the only feasible formulation for Prime(z) is obtained indirectly via the Normc(Comp(z))
path as shown in equation (12):
Prime(z) = Nat(z) - Normc(Comp(z)) - 1 - 1/z; .............................(12).
It is most probably impossible to generate a neat prime sequence for Prime(z) without using the
operator Normc( ). The reason is that Nat(z) is a normalised sequence whereas Comp(z) is an
unnormalised sequence. If Normc( ) is withheld, then the difference of Nat(z)-Comp(z) will yield
a mixed output sequence as shown in equation (13). This output sequence is correct for primes
but some composite numbers such as 4,9,25, and 49 are missing because these are exactly
subtracted. Nevertheless if the objective is to derive Prime(z), this is a measure of success
since this sequence can be extracted by sign differeniations. This is however awkward since
there is no suitable Maple function which could extract the positively signed algebraic terms.
There is an instrinsic function called type(integer,posint) which will return true if the integer is
positive but it does not work if an integer is associated with an order variable which is an
algebraic variable. It is interesting to note that Prime(z) is a normalised sequence. If any reader
could suggest how this could be overcome algebraically or by software, it would benefit the
author and other readers as well. Your contribuiton will be recorded in this webpage.
The method of finding Prime(z) via the Nat(z)-Comp(z) pathway is shown below:
Comp(z):=series(sum(1/(z^i*(z^i-1)),i=2..50),z=infinity,50);
Nat(z):=sort(sum(1/z^i,i=2..49));
Prime(z):=sort(Nat(z)-Comp(z));
..........................1.......1.........1.........1........1........1.......1.....1.........1.....3......1......1......1
Prime(z) := - O(---) + ---- + ---- + ---- - ---- + ---- - ---- - --- + --- - --- + --- - --- - ---
...........................50.......2.........3.......5........6........7........8.....10.....11.....12....13....14....15
..........................z........z.........z.........z........z.........z........z......z.........z.......z......z......z......z
..
........2......1......3......1......3......1......1......1......5......1........1......3......1......5......1......3
......- --- + --- - --- + --- - --- - --- - --- + --- - --- - --- - --- - --- + --- - --- + --- - ---
............16....17....18....19....20....21....22....23....24....26....27....28......29....30....31.....32
...........z......z......z......z......z......z......z.......z.......z.......z.......z.......z.......z.......z.......z.......z
.........1......1......1......6........1......1......1......5......1......5......1......3......3......1.......1......7
......- --- - --- - --- - --- + --- - --- - --- - --- + --- - --- + --- - --- - --- - --- + --- - ---
..........33....34....35......36....37....38....39....40....41....42....43.....44......45.....46....47....48
.........z......z......z........z.......z.......z......z.......z.......z.......z.......z......z........z.......z......z.......z........(13).
4. The Normc(Comp(z)) pathway to Prime(z)
This is the method currently used in sequence algebra. We show how Prime(z) is correctly
derived using the Normc(Comp(z)) pathway.
Normc(Comp(z)) :=
...1......1......1........1.......1.......1.......1.......1.......1........1.......1........1.......1.......1.......1
---- + --- + --- + ---- + ---- + --- + ---- + --- + --- + --- + --- + --- + --- + --- + ---
....9......25.....49......4.......6......10......8......12.....14......15.....16.....18.....20.....21.....22
..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.......1......1.......1......1......1
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...........24.....26.....27.....28.....30.....32.....33.....34.....35.....36.....38....39....40....42....44....45
...........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
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- .......................................(14).
...........46......48....50.....51....52.....54....55....56.....57.....58
...........z.......z......z.......z......z.......z......z.......z......z.......z
..................1......1..........1.......1.......1......1.......1.......1.......1.......1........1......1.......1......1.......1
Nat(z) := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + ---- + --- + --- + --- + --- + ---
...................4......6...........9......12.....14....15.....18....21.....20......8.......10....16.....22....24.....25
..................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.......1.......1......1......1......1
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
............26.....27....28......49....50....51....52.....54.....55.....56.....57.....58.....30....33.....34....35
............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.......1........1........1........1
.....+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---- + ---- + ---- + --- + ---
...........36....38.....39.....40....42.....44....45.....46.....48......32......3.......5.......7......11.....13
...........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
.....+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- ..............................................(15).
.........17.......19.....23....29....31.......37....41....43....47....53
.........z.........z.......z......z......z.........z.......z.......z.......z......z
Prime(z):= Nat(z) - Normc(Comp(z));
................1.......1........1.......1......1......1........1........1......1......1.......1......1.......1.......1........1
.........= ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
..................3......5.........7......11....13....17......19......23....29....31.....37....41.....43.....47......53
.................z......z........z........z.......z.......z........z.........z.......z......z.......z.......z.........z.......z......z
............................................................................................................................................(16).
If a probable primality test method is used we also get the same result. However the above
method is absolutely determinstic whereas the primality method may return false primes. The
primality method is shown in equation (19).
Prime(z):=sum(1/z^i*(isprime(i)-false)/(true-false),i=3..58);
....................1......1........1........1........1.......1.......1......1......1......1.......1.......1.......1.......1.......1
Prim(z) := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.....................3......5........7........11......13.....17.....19....23.....29.....31.....37.....41.....43....47....53
....................z......z........z..........z........z.......z........z......z.......z........z........z.......z.......z.......z......z
.................................................................................................................................................(17).
5. Conclusions
Sequence algebra analyses of holistic number sequences are based on the defintion of Nat(z) as
an axiom to be added into the existing axiomatic set in number theory. It is shown that because
curve fitting of discrete integers is based on the use of periodic divisibility sequences, it strength
is in the derivation of composite number sequences. Sequences of indivisibilities, especially the
prime sequence Prime(z), cannot be derived directly since there is no direct test for
indivisibilities. Therefore Prime(z) is derived indirectly via the Comp(z) or the Normc(Comp(z))
path. It is shown that both methods are workable although the the Nat(z)-Comp(z) method will
return a mixed sequence in which the primes are positively signed and the composites negatively
signed. This opens the way to extract Prime(z) from the mixed sequence. If the Nat(z)-
Normc(Comp(z)) method is used, a pure Prime(z) sequence is obtained without further filtering.
However this latter method is dependent on the availabity of an intrinsic function called Normc( ).
This function is currently not implemented in symbolic softwares. Everything having been said,
the author cautions that sequence algebraic formulations are more suitable for dialectic analyses
than algorithmic developments. He concedes that there are more efficient algorithms already in
existence for computations of prime numbers. However, Comp(z) and Prime(z) formulations are based on
fundamental principles and will deliver dialectic insight to its users. There is a lot of truth in the
saying: "Dialectic methods deliver insights whereas algorithmic methods delivers results".
6. Epilogue
There is one special property in sequence algebra which the author has not highlighted
before and that is whenever sequence algebraic analysis fails, it gives one the reason
why it fails. This is important since it means that one knows what is wrong and can
proceed to find remedies. This cannot be said of traditional algebra. Very often, when
sequence algebra fails, it does not fail abruptly. It still gives the correct answers with
a "catch", i.e., you have got to think hard how to extract the information from irrelevant
output. For example, it is possible to detect perfect numbers and amicable numbers in
sequence expansions [8]. For example even the Nat(z)-Comp(z) path gives the correct
primes but it points to the need of a new sign operator which can work on signed
algebraic terms. This is an advantage one seldom encounter in traditional algebra.
The reason is that sequence algebra is very visually intuitive. Visualisation and
intuition are not favoured by pure mathematicians. On the other hand, pure mathematicians'
insistence on pure abstractions is far from being scientific. If pure thoughts are based
on a material object called the brain, how abstract could it get? Since our physical being is autopoietic,
and since knowledge orginates from the brain, surely knowledge itself must be
autopoietic as reflected by the connections of neural nets in the brains. There were
flying lizard 200 millions year ago. Surely there must be something which we can
learn from Nature with its 2 billion years of autopoeitic learning. Mathematicians
should study and borrow from biological evolution, molecular biology, theoretic
physics and modern neurological science in the search for new axioms some of
which are not at all self-evident. In other words, what is wrong with treating
mathematics as an empirical science?
7. References
1. Burton M. Burton: Elementary Number Theory, Third Edition, WCB publishers, 1994, 17 to
48 to 50.
2. Huen Y.K.
3. Huen Y.K.
4. Huen Y.K.
5. Huen Y.K.
6. Huen Y.K.
7. Huen Y.K.
8. Huen Y.K.
9. Huen Y.K.
10. Huen Y.K.
11. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci.
Technol., 1994, VOL. 25, NO.6, pp 913-920.
12. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
13. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1996,
VOL.??, NO.?, ???-??? (In the press as proof paper mes 100421).
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