Tables Of XGS4- and XGS5-Primes
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: 11/12/97 )
Abstract
XGS2- and XGS3-primes have been reported in a previous paper [15,16]. This paper continues
the search in XGS4- and XGS-5 primes. The general trend observed is that the higher the
power n by which the prime sequence is raised, the rarer the occurrences of primes in the
numerator coefficients of the result sequence.
1. Introduction
Continuing with the search for higher ordered XGS, this paper reports findings on XGS4-
and XGS5-primes. These are industrial grade XGS-primes found by a probabilistic Lucas-
primality tests.
The general procedures for finding these primes are outlined as follows:
Step (1): Generate a finite prime sequence using the Maple V R 3's syntax as follows:
Primex:=sum(i^(isprime(i)-false)/(true-false),i=2..ub); ...........(1).
The value of upperbound ub is limited by the power n by which the sequence is raised and
the amount of RAMs in the Pentium. The rule is that only the first n integers are accurate
and suitable for primality testing. For example, if one raises a prime sequence with ub=1000
to the power n = 5, then one expects an output sequence with 4999 integer terms but only
the first 1000 integer terms are determinstic. This is because terms higher than n are still not
yet "fully-filled" and are thus inaccurate for primality tests. To increase the accurate range,
you must increase the value of n beyond 1000 but this is only possible if the PC does not run
out of memory.
Step(2): Raise the prime sequence to the power n and expand. Then take the first ub terms
for primality tests. To cut down unncessary testing, remove all even integers and those
ending in 5s. The number of odd integers left is thus greatly reduced.
Step (3): Suppose the list of odd integers is given by {x1,x2,..............,xn-1,xn}. To perform
primality tests on the list, using the following Maple codeline:
map(isprime,{x1,x2,.................,xn-1,xn}); .............(2).
If the list contains primes, the output will report {false,true}. If the list contains only composite
numbers, it will return {false}. Then one divides the list into smaller lists and retests and
removes all those sublists which report {false}. For those which report {false,true}, you will
have to remove one integer at a time from the head of the list and test until you get
{false}. As soon as you detect a {false}, then the previous integer removed will be a prime
which you can recover from the clipboard. You may still miss some primes if the list
contains more than one prime as only the last prime of the list will be detected correctly.
Than you will have to retest using the orginal list minus the newly found primes. This is done
until sublists are exhausted.
Given below are Tables 1 and 2 summarising the lists of odd integers and industrial
grade XGS-primes detected. It can be seen that the occurrences of these primes are quite
rare.
Table 1: XGS4- Odd integers with XGS4-Primes Included (in bold fonts)
----------------------------------------------------------------------------------------------
XGS4:={801, 15271, 119067, 1829157, 4987293, 23005739, 47788499, 137946779,
940290949, 2762408469, 4815242697, 6962394977, 23098398311, 60627870899,
105774424271, 141461893607, 177406157687, 347209498919, 516302579369,
959183451113, 1101556956793, 1223801556677, 1495773013931,
2151412460417, 2148111643821, 2214770885443, 2053098120423, 2025452871507,
1749188066447, 826263091813, 445315076551, 107401498081, 25594951569,
12169539336, 1568239201, 25594951569, 1568239201,60655618663, 106219012087,
143203535451, 295845643761, 1283109374521,1828845067151, 2188689376313,
2565260379429, 6538343505801, 8608018025657, 146455215112089, 16861171064953,
37189264634933, 46536890212599, 52625441249899, 84439734132041, 95436151873277,
138642167844913x,206857671109179, 232254947371029, 289107847114217,
321098742154629}
---------------------------------------------------------------------------------------------------------------------
Table 2: XGS5- Odd integers with XGS5-Primes Included (in bold fonts)
-------------------------------------------------------------------------------------------
XGS5:={6203, 128978901, 471860803, 236500024969 , 421414875531,
5756107078843, 302603461010451,1058903371609583, 3844384102261647,
5141656129016571, 41998632953667721, 60347987805889707,70019882468637909,
136943930914273341, 400304587862456193, 1004704228431369473,
1152137982471448847, 1288662462610842719}
------------------------------------------------------------------------------------------------------------------
2. Compared to Mersenne's and Fermat's Primes
Definiton of Mersenne Prime: When 2^n-1 is prime it is saide to be a Mersenne prime.
The Maple line for generating industrial grade Mersenne Primes is:
sum((2^i-1)/x^i*(isprime(2^i-1)-false)/(true-false),i=2..ub); .............(3).
Currently GIMPS sets a record of 2^2975221-1 as the largest found Mersenne Prime which
contains 895,932 decimal digits [17]. Mersenne Primes grows to astronomical magnitudes very
fast. Using equation (3), we show below the expansion of the first 12 terms.
Mp(z) :=
..3........7........31.....127....8191....131071.....524287...2147483647...2305843009213693951
---- + ---- + ---- + --- ...+ ---- ..+ ------ .. + ------ ..+ ---------- .. + -------------------
.....2.......3.........5........7........13.........17...........19................31............................61
..x........x.........x.........x.........x...........x..............x..................x..............................x
....618970019642690137449562111...162259276829213363391578010288127
+ --------------------------------------+ ---------------------------------
.................................89...........................................................107
................................x.............................................................x
...170141183460469231731687303715884105727
+ ------------------------------------------------------ .....................................................(4).
............................................127
..........................................x
Within the range 2^31 = 2147483648, there are 8 Mersenne primes whereas within this range
there is only one each of XGS4 and one XGS5. These show that at least in the initial
ranges, these XGS-primes are rarer than Mersenne primes.
Defintion of Fermat Prime: If Fn = 2^(2^n)+1, n>=0 is a prime, then Fn is a Fermat prime.
The Maple line for generating industrial grade Fermat Primes is:
sum((2^(2^i)-1)/x^i*(isprime(2^(2^i)-1)-false)/(true-false),i=0..ub);.............(4).
Fermat Primes are very rare and grow even faster than Mersenne Primes. Given below is
the expansions of the first five Fermat primes, i.e., from Fo to F4, using equation (4) for i from 0 to 13.
..............................17....257...65537
Fp(z) := 3 + 5/x + ---- + --- + ----- ......................................(5).
.................................2......3........4
...............................x.......x........x
It is not known whether there is at least one Fermat prime beyond F4. The best"guess" is
that Fermat numbers Fn > F4 are composite. This make Fermat primes very rare indeed
and makes XGS4 and XGS5 falling somewhere between Mersenne and Fermat primes in
scarcity. This remark is only true for the initial range as the search had not be conducted
exhaustively. Furthermore the XGS-primes reported in this paper are industrial grade
primes.
3. Summary
This concludes our short excursions into XGS-sequences and primes from XGS2 to XGS5
before we move onto something else. It is hoped that those interested will take these up for more
detailed investigations.
4. Reference:
[Comments: Papers included in this section are relevant for background readings
but not all are directly referenced in the main text. You can hyperlink to most of these
papers from within this reference section.
========================================================
1. Huen Y.K.:
A matrix map for primes and nonprimes, Int. J. Math.Educ.Sci.Technol., 1994,
Vol.25, No.6, pp 913 - 920.
=======================================================
2. Huen Y.K.:
Visual algebra and its applications, Int. J. Math. Educ. Sci. Technol., 1997,
Vol.28, No.3, 333-344.
=======================================================
3. A Simple Introduction To Sequence
Algebra - by Huen Y.K.
(date release: 15.3.97) (38 KBytes, 11*A4 pages).
========================================================
4. Methods Of Developing Sequence
Algebraic Formulations For Comp(z) and Prime(z) - by Huen Y.K. (date released : 20.6.97) (36.8 KBytes, 10*A4s).
========================================================
5. Lemmata, Corollaries, And
Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).
========================================================
6. Improved Formulations For Comp(z)
and Prime(z)
- by Huen Y.K. (date released : 16.9.97) (17 KBytes ).
========================================================
7. Detecting False Reports
in Primality Tests By The Oddcomp(z) Method.
- by Huen Y.K. (date released : 18.9.97, Revised 20/9) (26 KBytes ).
========================================================
8. The Throwing Power Of
Oddcomp(z).
- by Huen Y.K. (date released : 24.9.97 ) (15 Kbytes).
========================================================
9. Sequence Algebraic
Approach To Prime Number Theorem
- by Huen Y.K. (date released : 28.9.97 ) (21 Kbytes).
========================================================
10. Generating Functions -
Closed Forms vs Open Forms
- by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).
========================================================
11. Generating Large
Odd Composite With Two Prime Factors
- by Huen Y.K. (date released : 3.10.97 ) (13.5 Kbytes).
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12. In Search Of Counter-
Examples In Maple's Isprime Function.
- by Huen Y.K. (date released : 4.10.97 ) (18 Kbytes).
========================================================
13. A Sequence Algebraist's
View Of Lehmann's Primality Test
- by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).
========================================================
14. On Odd(z), Oddcomp(z),
Seq1(z) and Seq2(z)
- by Huen Y.K. (date released : 10.10.97 ) (17 Kbytes).
========================================================
15.In Search Of Primes From
The XGS Sequence
- by Huen Y.K. (date released : 10.12.97 ) (26 Kbytes).
=========================================================
16. Tables Of XGS4- and XGS5-Primes
- by Huen Y.K. (date released : 11.12.97 ) (26 Kbytes).
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17. GIMPS Sets Another Record! ....
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