Sequence Algebraic Approach To Prime Number Theorem
by
Huen Y.K.
CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg
(A short communication - 1st released: 28/9/97 )
Abstract
Although the celebrated Prime Number Theorem predicts fairly accurately the number of
primes less than a given number n by the expression n/(log n), this is only a probabilistic
prediction [1,2]. The sequence algebraic formulation can do better than this as the
determination of Oddcomp(z) is deterministic. Once the sequences of odd integers and of
odd composite integers are
found, the sequence of primes can be found deterministically by the difference between them
up to the number n. Sequence algebraists tackle the Prime Number Theorem from a totally
different axiomatic basis. No connection at present can be found between probablistic Prime
Number Theorem and sequence algebraic Prime Number Theorem.
1. Introduction
This paper starts by comparing two sequence algebraic formulations for predicting composite
integers given by equations (1) and (2). Equation (1) is by now the well known equation of
divisibles which predicts deterministically the sequence of integers which includes both even
and odd composites [8]. Equation (2) predicts only the odd composite integer sequence [7].
......................................upperbound
..........................................-----
...........................................\............1
...........................Comp(z):= )..... ----------- ...............................................(1).
.........................................../............i....i
..........................................----- ....z..(z - 1)
..........................................i = 2
..............................upperbound
..................................-----
...................................\........................1
.............Oddcomp(z):= )......-------------------------- .......................................(2).
.................................../...........(2 i + 1)...(4 i + 2)
..................................-----....z............(z..............- 1)
..................................i = 1
For the prediction of the Prime(z) sequence, the use of equation (2) has been found to be far
more efficient than equation (1) for the following reasons:
(i) Odd members of Prime(z) belong to a subset of Oddcomp(z). Since the equation of
divisibles given by equation (1) does not discriminate between odd and even composite
numbers, it wastes computing resources if one uses it to derive Prime(z).
(ii) The throwing power of sequence algebraic generating function is defined by the ability of
such a function to predict the number of contiguous terms correctly up to n when the number
of divisors used is less than n. The throwing power of Comp(z) is given by k = 2 but that of
Oddcomp(z) is given by k = 172. This means that the throwing power of the latter is about
85 times better than the former [7].
The deterministic formulation for Prime(z) is derived as shown in equation (3). Note that the
even prime of 2 has to be added separately.
.................................................................upperbound
.....................................................................-----
...........................z.................1.......................\.....................1
Prime(z) := -------------- + ------ - Normc [....)......---------------------------.....] ........(3).
.....................z^2(z - 1).........z^2..................../...........(2 i + 1) (4 i + 2)
.....................................................................----- ...z............(z.............- 1)
.....................................................................i = 1
2. Verification
Since Maple V R 3 does not provide the newly defined function called Normc( ) which
operates by reducing all numerator coefficients of terms in Oddcomp(z) to unity, Normc( )
will be performed manually in this test. The general Maple's program line for computing
Oddcomp(z) is computed first using equation (4). The steps are described below:
Step(1): Oddcomp(z) is computed based on the general Maple line shown in equation (4) by
putting k = 1.
Oddcomp(z):=sort(series(sum(1/(z^(2*i+1)*(z^(2*(2*i+1))-1)),i=1..upperbound),
z=infinity,k*upperbound); ...........................................................................(4).
The above equation is tested for a finite expansion for i = 1..100 as shown in equation (5).
Oddcomp(z):=sort(series(sum(1/(z^(2*i+1)*(z^(2*(2*i+1))-1)),i=1..100),z=infinity,100));
...................................................(5).
The resultant odd composite sequence is shown in equation (6).
.............................1.........1.....2......2......1.....2.......2......2......2.......4........1.......2.......2.......2
Oddcomp(z):=O(----) +----+--- +--- +--- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.............................105.......9.....15....21....25...27....33.....35....39.....45......49.....51.....55.....57
............................z..........z.......z......z.......z.....z.......z.......z.......z.......z.........z.......z.......z........z
.
.........4.......2.......2......4......2.......3.......2.......2.......2.......2.......2......4
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- ..........................(6).
............63....65.....69.....75.....77.....81....85.....87.....91.....93.....95.....99
..........z.......z........z.......z.......z........z.......z.......z........z........z.......z.......z
Step(2): The numerator coefficients of Oddcomp(z) in equation (6) are reduced to unity by
applying the Normc( ) operation resulting in equation (7).
Normc(Oddcomp(z)):=
...................1......1......1.......1.......1.......1......1......1.......1.......1.......1.......1.......1......1......1
................ ---- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
....................9......15.....21.....25.....27.....33....35....39.....45.....49.....51.....55.....57.....63.....65
...................z......z.......z........z.......z........z.......z......z.......z........z........z........z.......z.......z......z
...........1......1......1........1.......1.......1......1.......1.......1
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- .........................................(7).
.............69....75....77......81.....85.....87....91....95.....99
...........z.......z......z.........z.......z........z......z.......z.......z
The odd integer sequence can be found by equation (8) and the even integer of 2 added
subsequently.
Odd(z):=sort(series(z/(z^2*(z^2-1)),z=infinity,100));
......................1...........1........1.......1........1........1.......1.......1.......1......1......1.......1.......1......1
Odd(z) := O(----) + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.......................101........3........5.......7........9.......11.....13......15.....17....19.....21....23....25....27
......................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
.......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
............29.....31.....33.....35.....37.....39....41....43.....45.....47.....49.....51.....53.....55.....57....59
............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......1
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
.............61....63......65.....67.....69.....71....73....75.....77....79.....81.....83.....85.....87....89....91
.............z.......z.......z.......z........z.......z.......z.......z.......z.......z.......z........z.......z.......z.......z.......z
..........1.......1.......1.......1
......+ --- + --- + --- + --- ........................................................................(8).
...........93.....95.....97.....99
..........z.......z.......z.......z
The prime sequence including the even prime of 2 is shown in equation (9):
Prime(z):=1/z^2 + sort(Odd(z)-Normc(Oddcomp(z));
......1........1.......1......1......1......1.....1......1......1.......1.......1......1.......1......1......1......1........1
O(----)+----+----+----+----+--- +--- +--- +--- + --- + --- + --- + --- + --- + --- + --- + ---
.......101.....2.......3......5......7.....11....13...17.....19....23.....29.....31....37.....41....43.....45.....47
......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
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- ...............................(9).
...........53......59....61....67....71.....73.....79.....83.....89.....93.....97
..........z.........z......z.......z......z........z.......z.......z........z.......z.......z
Step(3): To count the number of odd primes in this interval, we apply a first order integration
as follows:
sort(series(1/(z-1)*(Odd-Normc(Oddcomp(z)),z=infinity,100));
.......1.........1........1........2........2........3.......3........3........3.......4.......4.......5......5......5.......5
O(----) + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- + ---
......100........4 .......5........6.........7.......8.......9........10.....11......12.....13.....14.....15.....16....17
..........z........z.......z........z.........z........z.........z........z........z........z........z........z......z........z......z
..........6.......6.......7........7......7......7.......8.......8.......8.......8.......8......8......9......9.....10....10
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...........18......19....20.....21......22.....23....24....25.....26.....27....28.....29......30.....31....32....33
..........z........z.......z.......z........z.......z.......z.......z......z.........z.......z.......z........z........z......z......z
.........10.....10.....10.....10......11....11....11....11....12.....12.....13.....13.....12....12....13.....13
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...........34....35....36.....37......38....39.....40.....41....42.....43....44......45......46....47....48....49
..........z........z.......z.......z........z.......z.......z.......z......z.........z.......z.......z........z........z......z......z
..........13....13.....13.....13......14....14...14.....14.....14.....14.....15....15.....16.....16....16....16
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...........50.....51....52.....53.....54....55.....56.....57....58.....59....60.......61.....62.....63....64....65
..........z........z.......z.......z........z.......z.......z.......z......z.........z.......z.......z........z........z......z......z
.........16.....16.....17.....17......17....17....18....18......19....19.....19....19.....19.....19....20....20
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
...........66.....67....68.....69.....70....71.....72....73.....74.....75.....76......77.....78.....79....80....81
..........z........z.......z.......z........z.......z.......z.......z......z.........z.......z.......z........z........z......z......z
..........20.....20.....21.....21.....21....21....21....21......22....22.....22....22.....23....23.....23....23
......+ --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + ---
............82.....83....84.....85.....86....87.....88....89.....90.....91.....92.....93.....94.....95....96....97
..........z........z.......z.......z........z.......z.......z.......z......z.........z.......z.......z........z........z........z......z
.........24....24
......+ --- + --- ............................................................................................(10).
...........98....99
..........z.....z
The number of odd primes is given by the numerator of the last term as 24. Add to it the
even prime of 2 gives a total of 25.
Confirmation: This can be confirmed using Maple's isprime function which returns true for a
prime and false otherwise as shown in equation (11).
sum(x^isprime(i),i=2..100);
................................................true..........false
........................................25 x ......+ 74 x .................................................(11).
3. Conclusions
The deterministic and general formulation to predict Prime(z) up to n = 2*upperbound-1 can
be written down as shown in equation (12). It is understood that the numerator coefficient of
the last term will measure the number of primes in the sequence.
.............................................................upperbound
.....................................................................-----
.............................1.............1........................\ ...................1
.........Prime(z):= -------+ ---------- - Normc[.)......--------------------.........] .......(12).
...............................2.............2....................../.........(2 i + 1)....(4 i + 2)
.............................z.........z (z - 1).................----- ...z.............(z............ - 1)
.................................................................i = 1
Thus after series expansion of Prime(z) in equation (12), one could count the number of
primes by applying a first order integration of 1/(z-1) to this sequence which upon another
series expansion will give the total count of primes in the last term at (2*upperbound-1).
Thus in accordance with the conventional notation for Prime Number Theorem we have
sigma(n)/n = 1/(2*upperbound-1)*series(1/(z-1)*Prime(z),z=infinity,2*upperbound-1);
This is a deterministic formulation compared against the statistical formulation given by the
celebrated Prime Number Theorem originally discovered by Gauss which was subsequently
improved by Hadarmard and Poussin [1]. There is no apparent connection between the
sequence algebraic formulation with the probablistic formulation. The author fails to detect
where Riemann's zeta function could arise in the sequence algebraic formulation. After all
Riemann's hypothesis is still a hypothesis albeit that many mathematicians accept it as
true.
An interesting observation in equation (12) is that it is based on the difference of two periodic
sequences although the Oddcomp(z) expression under Normc( ) operator has compound
periodicities. This might explain why it is so difficult to detect periodicities in the prime
sequence.
Algorithmically, the formulation is not expected to be as efficient as the probabilistic primality
tests but because it is deterministic, it serves a different function. Since primes are all
deterministically predicted, it can be used to validate probable primes found by primality
methods. The Oddcomp(z) formulation can be used to weed out probable primes efficiently
even in the very large end of the integer number sequence because once a number is found
to be an odd composite, there is a 100% confidence that it will never be a prime. However, if
Oddcomp(z) is not expanded starting from i = 1, then some small divisors will be missed and
thus no deterministic conclusion could be arrived at without the presence of an
odd composite integer.
Repeatedly, the author encountered number sequences which, contrary to appearance, turned
out to be periodic and can be predicted by sequence algebraic formulations [5 to 11]. So far
he has succeeded in finding a periodic formulation for Pi and some irrational numbers in
sequence algebraic formats [12].
4. Reference:
1. Davis P.J.and Hersh R.: The Mathematical Experience, Penguin Book, 1981, Great
Britain, pp 209 to 216.
2. Burton D.M.: Elementary Number Theory (Third Edition), WCB Publishers, 1989, Dubuque, pp 326 to 332.
3. Huen Y.K.: Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z). URL site: http://web.singnet.com.sg
/~huens/ .
4. Huen Y.K.: Improved Formulations For Comp(z) and
Prime(z). URL site: http://web.singnet.com.sg/~huens/ .
5. Huen Y.K. Lemmata, Corollaries, And Theorems In Sequence Order Analysis. URL site:
http://web.singnet.com.sg/~huens/ .
6. Huen Y.K.: Detecting False Reports In Primality Tests By The Oddcomp(z) Method, URL
site: http://web.singnet.com.sg/~huens/ .
7. Huren Y.K.: The Throwing Power Of Oddcomp(z), URL site: http://web.singnet.com.sg/
~huens/
8. 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.
9. Huen Y.K.:
10. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1997,
VOL.28 NO.3, 333-344.
11. Huen Y.K.: Twin-primes revisited. Int. J. Math. Edu. Sci. Technol. 1997, Vol.???. No.
???-???. ( the the press as proof paper mes 100488).
12. Huen Y.K. :Is Pie Periodic? INT.J.Math.Educ.Sci. Technol., 1997, Vol??, No.?,???..???,
(In the press as proof paper mes100495).
======================== END OF PAPER ======================