1. Introduction
Readers who are only interested in the challenge problem are requested to proceed directly to section (4) of this
paper.
Given a finite sequence of integers, derive a deterministic, closed global formulation which can predict all
subsequent members of the sequence. Some sequence problems are posted in A.T. & T. Integer Research
Website but those do not require finding any closed forms [3].
With the development of sequence algebra since 1994, finding determinstic formulations for Mersenne primes
and Fermat primes has become quite straightforward provided either mixed mode arithmetic or the use of
Normc( ) operator is allowed. This paper explains how exotic composite sequences can be developed which can rival the difficulties encountered in the prediction of
prime sequences such as Mersenne prime and Fermat prime sequences. Note that the author refers to
these as sequences rather than as Mersenne and Fermat numbers [16]. For demonstration of the
algebraic techniques involved, the author will describe how these two formulations are developed. Then attention
will be shifted to the development of exotic composite sequences. The latter will have more recreational values
since the common starting point is the general equation of divisibles which by now should be familiar to visitors
to this URL-site.
2. Mersenne Primes And Fermat Primes
(i) Explicit Formulation For Mersenne Prime Sequence: Merzprime(z)
Even now number theoretic hunters tend to concentrate on individual Mersenne primes and Fermat primes
instead of their holistic sequences. There is a Mersenne Prime Webpage hosted by Chris K. Caldwell which
reports the latest findings on Mersenne primes and Perfect numbers amongst other interesting historical facts [2].
Until 1996, no determinstic algebraic formulations have been developed for Mersenne primes or Fermat primes[17].
There exists a defintion for Mersenne prime which offers meagre help to those in the Mersenne hunt. It is as
much help to a blue whale hunter by telling him that this is the largest creature on planet earth and that it is
somewhere in the vast ocean.
.............Definition: When 2^n -1 is prime, it is said to be a Mersenne prime.
If a coincidence between a prime in the Mersenne prime sequence and one in the natural prime sequence is
defined as a hit, then the hit rate for Mersenne primes is very low indeed and for Fermat primes even lower. This
can be easily demonstrated by a simple Maple program line with embedded primality tests. Such primality tests
are probablistic but so far no discovery of false primes have been reported. Primality tests are convenient but
sequence algebraists have a superior weapon -- a determinstic global formulation for the prime sequence called
Prime(z) [13]. Equation (1) shows the Laurent series of the first 100 terms of a Mersenne number sequence
generated using primality tests on the integer i. All Mersenne primes are displayed in bold fonts.
Even amongst the first 100 integers, the hit rate is already quite low but we know that intervals between
successive Mersenne primes grow very fast with large integers. This explains why it is so difficult to break world
records in the Mersenne hunt [2]. This is because determinstic primality tests on large integers will get
increasingly arduous and the rarity of large Mersenne primes probably turns this into a game of patience. It looks
as if the Mersenne hunt will in future be the reserve of big game hunters, i.e., those in possession of mainframes.
The author recommends sequence algebraic games since these do not require immense number crunching
power.
Mersenneprime(z):=sum(x^i/z^(2^i-1)*(isprime(i)-false)/(true-false),i=2..100);
......................................2.......3.......5.......7.........11.........13..........17...........19..............23
....................................x.......x.......x.......x......... x...........x...........x..............x...............x
Mersennenumb(z) := ---- + ---- + --- + ---- + ------ + ------ + -------- + --------- + ----------
......................................3........7......31....127....2047.....8191......131071....524287.....8388607
....................................z........z.......z.......z.........z............z............z..............z...............z
...........29.............31................37.......................41............................43.............................47
.........x...............x...................x.........................x..............................x...............................x
+ -----------+ -------------+ ---------------+ ------------------+ -----------------+ ----------
...536870911..2147483647..137438953471..2199023255551..8796093022207..140737488355327
z....................z.....................z........................z............................z..........................z
...............53..........................59.....................................61.................................67
..............x...........................x.......................................x...................................x
+ -------------------+ -----------------------+ --------------------------+ -----------------
9007199254740991 576460752303423487 2305843009213693951 147573952589676412927
z...............................z...................................z.......................................z
.....................71..........................................73........................................79
...................x............................................x..........................................x
+ --------------------------------- + --------------------------------- + -------------------------
...........2361183241434822606847....9444732965739290427391....604462909807314587353087
..........z.............................................z..............................................z
......................83.............................................89
....................x...............................................x
+ ------------------------------- + ---------------------------------
..9671406556917033397649407....618970019642690137449562111
z...................................................z
.....................97
...................x
+ -------------------------------
....158456325028528675187087900671 ...................................................................(1).
..z
We proceed to develop a determinstic formulation for Mersenne primes as follows:
Nat(z):=series(z/(z-1),z=infinity,10);
..........................................1..........1.......1........1.......1.......1.........1.........1............1
..........Nat(z) := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---) .......(2)
............................................2..........3.......4........5........6.......7........8..........9..........10
..........................................z...........z.......z........z.........z.......z........z..........z.............z
Comp(z):=series(sum(1/(z^i*(z^i-1)),i=2..10),z=infinity,10);
................................................1.......2.........2.........1............1
............................Comp(z) := ---- + ---- + ---- + ---- + O(---) ...................................(3).
..................................................4.......6.........8.........9...........10
.................................................z.......z.........z.........z...........z
One already notices that Nat(z) is normalised but Comp(z) is not. To proceed we need to normalise Comp(z) as
follows:
.................................................1..........1........1........1...........1
.................Normc(Comp(z)) := ---- + ---- + ---- + ---- + O(---) ...................................(4).
...................................................4..........6........8........9..........10
.................................................z...........z........z.........z..........z
Now Prime(z) can be derived by the difference between Nat(z) and Normc(Comp(z)) as follows:
Prime(z):= Nat(z) - Normc(Comp(z)) - 1 - 1/z
.............................1......1........1........1........1.......1.......1.......1.......1.......1......1
........Prime(z) := ---- + ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + --- ................(5).
...............................2......3........5........7.......11......13......17.....19.....23.....29.....31
.............................z.......z........z.........z........z.........z........z.......z.......z........z.......z
Merznumb(z):=sum(1/z^(2^i-1),i=2..5);
..................................................1........1......1........1
......................Merznumb(z) := ---- + ---- + --- + ----... ................................(6).
....................................................3........7......15......31
..................................................z.........z.......z.......z
By applying logic AND between equations (5) and (6) Mersenne primes can be filtered out as shown in equation
(7). Even though a short example is shown here, this is a global formulation for Mersenne primes.
Merzprime(z):= [AND Prime(z) Merznumb(z)] :=
...................................................1........1.........1
.......................Merzprime(z) := ---- + ---- + ------- +... ................................(7).
.....................................................3........7........31
...................................................z.........z........z
(ii) Explicit Formulation For Fermat prime sequence: Fermatprime(z)
Fermat primes are harder to compute since these grow big even faster and the intervals between successive
Fermat primes are even more nonlinear. Nevertheless the techniques developed for Mersenne prime sequence
can be applied as follows:
Prime(z):= Nat(z) - Normc(Comp(z)) - 1 - 1/z
...............................1........1.......1......1.......1.......1.........1
.........Prime(z) := ---- + ---- + ---- + ---- + --- + --- + --- + .....
.................................2.........3.......5......7......11......13.......17
...............................z..........z........z.......z.......z........z.........z
...............................1........................1....................................1
............................. ---- +...... + -------------- + .........+ ---------------
.................................257...................65537..........................4294967207 ............(8).
...............................z.......................z...................................z
Fermatnumb(z):=sum(1/z^(2^(2^i)+1),i=2..5);
.......................................................1........1...........1.................1
.........................Fermatnumb(z) := ----- + ------ + -------- + -------------
........................................................17......257........65537.........4294967297 ........(9).
.......................................................z........z............z.................z
..
By applying logic AND between equations (8) and (9) the Mersenne primes will be filtered out as shown in
equation (7). Even though a short example is shown here, equation (9) is a global deterministic formulation for
Fermat primes since it does not make use of primality tests.
Fermatprime(z):= [AND Prime(z) Fermatnumb(z)] :=
.......................................................1.............1.............1
........................Fermatprime(z) := ------ + ------- + --------- ..................(10).
........................................................17..........257...........65537
.......................................................z............z..............z
It can be seen that Fermatprimes(z) gets exceedingly rare as integers move into the very large end.
3. Exotic Composite Sequences
All exotic composite sequences has its starting point from the generalised equation of divisibles given by
equation (11):
..................................................................1
...................................Gencomp(z) := ----------------- .........................................(11).
.............................................................f(i).....f(i)
............................................................z.....(z.... - 1)
In the case of the natural composite sequence Comp(z), f(i) = i is linear. All one needs to do is to find functions
of f(i) which are nonlinear and the magnitudes of which grow very fast. The condition which is imposed in the
challenge problem posed in this paper is that f(i) must compute to positive integers. A demonstration examples
is given here in equation (12). The present problem is still quite mild since intervals are purposedly kept small.
Gencomp(z):= series(sum(1/(z^(i^2+i+1)*(z^(i^2+i+1)-1)),i=2..90),z=infinity,90);
..........................1.......1.......1......1......1.......1.......2......1......1......1......1.......2.......1......1......1
Gencomp(z) := --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + --- + -- + --
..........................14.....21....26.....28....35.....39......42.....49.....52....56....62.....63.....65....70....77
.........................z.......z.......z........z.......z........z........z......z.......z......z......z.......z.......z.......z.......z
.............................1.......2.......1...........1
.........................+ --- + --- + --- + O(---) .......................................................... (12).
..............................78......84......86........91
.............................z.......z.........z..........z
Thus the demo-problem posed will be as a normalised sequence given by the integer set shown in equation (13).
The challenge will be to find the closed formulation f(i) for this composite sequence. Of course the correct
answer is already given in equation (12) as i^2+i+1.
{14 21 26 28 35 39 42 49 52 56 62 63 65 70 77 78 84 86 91} ......(13).
4. The Challenge Problem Called Comp1_97(z)
The announcement is made of a challenge problem called Comp1_97(z) in this webpage. This paper can be
found under section (8) of this webpage. All subsequent challenge problem for composite sequences will bear
the label Compn_97(z) for the year 1997. Here is the first challenge problem:
Comp1_97(z): Find the close sequence algebraic formulation f(i) for the following composite integer set.
Note that Comp1_97(z) is defined to take the same upperbound ub for both the parameters i and z.
Comp1_97(z):= series(sum(1/(z^f(i)*(z^f(i)-1)),i=2..ub),z=infinity,ub);
............{ 22 33 44 55 62 66 77 88 93 99 110 121 }
Please just email the f(i) expression to the author at address below. If your
submission is received you will get an acknowledgement by return email. If you do not hear from the organiser,
please resubmit another one within 24 hours.
.............huens@singnet.com.sg or huens@mbox3.singnet.com.sg
The closing date for the challenge is at 12.00 noon Singapore time on 31.8.97, i.e. August the 31st of 1997.
The first correct submission received will get a nice Certificate of Achievement recording your findings which
you could display in your living room or your workplace. The second and third correct answers will also get
Certificates of Merit.
Final results will be announced in this webpage.
6. Conclusions
This paper is designed to introduce a new recreational game in composite number
sequences in addition to the original game of the grand search for full Goldbach's
Sequence which is still in progress. The objective is to introduce games which are more algebraic
than numerical so that these are accessible to most PC owners.
7. References
1. Burton M.D. :Elementary Number Theory, Third Edition, WCB Publisher,
1994, pp 203 to 226.
2. Chris K. Caldwell
3. AT & T Integer Sequences Research
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.
12. Huen Y.K.
13. Huen Y.K.
14. Huen Y.K.
15. 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.
16. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ.
Sci. Technol., 1996, VOL.27, NO. 5, 685-691.
17. 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).
18. Huen Y.K.: Twin primes revisited: INT. J. Math. Educ. Sci. Technol., 1997, VOL.??,NO.?, ???-???. (In the
press as proof paper mes 100488).
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