Generating Functions For Regular Prime k-tuple Sequences

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 - 3/3/98)

Abstract

The definition of a regular prime k-tuple in this paper differs from that used in Tony Forbes' homepage on prime k-tuplet [1]. There Tony Forbes and fellow workers are looking for the largest known examples of certain types of dense clusters of prime numbers. A regular prime k-tuple sequence is defined as a finite sequence of primes with fixed intervals of k units. Such sequences do not necessarily contain successive primes and numerical investigations carried out so far show that these are finite. In this paper we are looking for the first occurrence of the longest prime k-tuple in a sequence with fixed intervals of k units. The definition would have classified twinprimes and triplet-prime under regular 2-tuples as these have regular intervals of 2 units and thus belong to the smallest member of the family. The objective of this paper is to carry out systematic but limited investigations to uncover characteristic properties common to regular prime k-tuples.


1. Introduction

An ordered set of odd integers (p0,pk,...,pjk) with constant intervals between successive members can be represented in sequence algebra by a finite Laurent series as shown in equation (1) where j*k represents j integer multiples of interval of k units measured from the first member p0. If these integers are primes, we call this a regular prime k-tuple sequence to distinguish it from prime k-tuplets referred to in Tony Forbes' homepage [1]. Tony Forbes' k-tuplets represent the densest stretch of successive primes within a specified integer range whereas a regular prime k-tuple represents the longest contiguous sequence of primes with regular intervals of k units. In k-tuples, primes are not necessarily successive primes although some successive primes do occur within these sequences. All the k-tuple sequences found so far are of finite lengths but there is no proof that out there somewhere there may exist some with infinite lengths although from knowledge of the nonlinear distributions of primes, these may be very unlikely. This paper assumes that such sequences are of finite lengths and will only report the first occurrence of the longest k-tuple found for each value of k starting from 1. Most k-tuples are distinct but there are some with multiple occurrences with interlaced arrangements where the first occurrence will also be reported. Note that intervals between successive primes are given by 2*k since such successive interval must be even-valued if begin a sequence with an odd integer.

k_tuple:sum(sum(sum(1/z^(2*i-1+2*j*k)*MMF,j=0..uj),i=1..ui),k=k1..k2);

where the expression 2*i-1 ensures that the first integer in the sequence is odd, and 2*j*k ensures that all subsequent intervals measured from the first integer are even, and MMF is the acronym for a prime filter represented by the product of primality predicates which return the numerical value of 1 if all terms tested are primes and 0 otherwise [2]. To keep computations simple, we only compute for a fixed value of k, i.e., k = k1 = k2.


2. Generating Functions For Regular Prime k-tuples

The general generating function used in the computations is given by equation (2). Note that the second bracketted expression represents the MMF included in equation (1).

Maple: k_tuple:=sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..uj)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..uj),i=1..ui),k=k1..k1);

Macsyma: k_tuple:sum(sum(sum(1/z^(2*i-1+2*j*k),j,0,uj)*product((primep(2*i-1+2*j*k)-false)/(true-false),j,0,uj),i,1,ui),k,k1,k1);

........................./ / 1 \(uj + 1)...........................\
................... ui | |-----|.....................................| /....uj.........................................\
.................---- | |....2 |...2(2*k1).........(2*k1)...| | ------'.....................................|
..................\.....| \ z.../ z..................z.................| |'..| | (isprime(2i-1+2jk1) - false)|
k_tuple :=...)...|- -------------- + -------------| |...| | ----------------------.........|.......(2).
................../.....|...2i-1........2k1....2i-1........2k1| |...| |..........true - false.................|
.................---- \ z (- 1 + z )........z...(- 1 + z ).../ |...| |...........................................|
.................i = 1..................................................\ j = 0......................................./


3. Computations Of Specific Regular Prime k-tuples

Regular prime 1-tuple to 50-tuple are computed using equation (2) and reported in Table 1. Beyond 50-tuples, only those with k-tuples equal to sextuplets and above are reported. Note that a probabilistic primality test called isprime is used which would relegate these primes to industrial grade but the chance of a false report is extremely rare and can be easily checked within the low range of i from 0 to 10000.

Computations are based on Equation (2) using program lines in Maple V R 3. Each tuple is computed with a fixed value of interval multiplier k. In a few cases, these occur in multiplicities and only the first occurrence will be reported.

Table 1 -- Regular Prime k-tuples from k= 1 to 50
(1) Prime 1-tuples
This is the only problem worked in full. Note that we call this sequence the 1-tuple which means that the intervals between successive primes are of 2 units. The first occurrence is a prime triplet of (3,5,7).

    1_tuple:=sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false) /(true-false),j=0..2),i=1..10000),k=1..1);

    ...................1........1........1
    1_tuple := ---- + ---- + ---- ...........................(3a).
    .....................3........5........7
    ...................z........z.........z

    Note that in the above Maple program line, prime tests are taken over three odd integers at a time. This reports the occurrence of the first and only triplet prime set {3,5,7}. In the next line, we take two odd integers at a time which returns both the triplet and subsequent prime twins. As we are only interested in the first occurrence of the longest 1-tuple, in this case, we will only report the one and only prime triplet. This remark applies to all subsequent findings.

    1_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/ (true-false),j=0..1),i=1..20),k=1..1));

    ..................1.........2.......1.......1.......1.......1......1.......1......1
    2_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + ---...........(3b).
    ....................3.........5.......7.......11....13.....17....19.....29.....31
    ..................z.........z........z........z.......z.......z.......z.......z.......z

(2) Prime 2-tuples
The first occurrence is a triplet given by (3,7,11).

    2_tuple:=sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..10000),k=2..2);

    ..................1........1........1
    2_tuple := ---- + ---- + ---................................................(4).
    ....................3........7.......11
    ..................z........z........z

(3) Prime 3-tuples
The first occurrence is a quintuplet given by (5,11,17,23,29).

3_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..4)*product((isprime(2*i-1+2*j*k)-false)/ (true-false),j=0..4),i=1..10000),k=3..3));

    ..................1........1......1......1.......1
    3_tuple := ---- + --- + --- + --- + ----................................................(5).
    .....................5......11.....17.....23.....29
    ..................z........z.......z.......z........z

(4) Prime 4-tuples
The first occurrence is a triplet given by (3,11,19).

    4_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..10000),k=4..4));

    ..................1........1.......1
    4_tuple := ---- + --- + ---..................................................(6).
    ....................3.......11.....19
    ..................z........z........z

(5) Prime 5-tuples
The first occurrence is a triplet given by (3,13,23).

    5_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..10000),k=5..5));

    ..................1........1.......1
    5_tuple := ---- + --- + ----....................................(7).
    ....................3.......13.....23
    ..................z........z........z

(6) Prime 6-tuples
The first occurrence is a quintuplet given by (5,17,29,41,53).

    6_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..4)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..4),i=1..30000),k=6..6));

    ...................1......1.......1......1.......1
    6_tuple := ---- + --- + --- + --- + ---....................................(8).
    .....................5.....17......29.....41.....53
    ...................z......z........z.......z.......z

(7) Prime 7-tuples
The first occurrence is a triplet given by (3,17,31).

    7_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..100),k=7..7));

    ...................1......1.......1
    7_tuple := ---- + --- + ---.....................................................(9).
    .....................3.....17......31
    ...................z......z........z

(8) Prime 8-tuples
Note that the 8-tuples are all twins with fixed intervals of 16 units are interlaced at least in the small range. Due to existence of multiple occurrences, one cannot refer to any as the longest tuple. The first prime twin is (3,19).

    8_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..10000),k=8..8));

    ..................1.........1.......1........1.......1......1......1.......1......1.......1
    8_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + ---+ ----...........(10a).
    ....................3.........7......13......19.....23.....29....31.....37....47.....53
    ..................z.........z........z........z.......z.......z........z.......z.......z.......z

    Given below are computations of 8_tuples at a high range of i from 2^150 to 2^150+10000 which show that there still exist prime twins but at a greatly reduced density and that there is no more interlacing amongst the prime pairs. This is to be expected since the average intervals between primes increase with magnitudes of primes.

    ..................1
    8_tuple := ---------------------------------------------------------------
    ....................2854495385411919762116571938898990272765497057
    ..................z

    ....1
    + ---------------------------------------------------------------
    ......2854495385411919762116571938898990272765497073
    ....z

    ....1
    + ---------------------------------------------------------------
    ......2854495385411919762116571938898990272765503957
    ....z

    ....1
    + ---------------------------------------------------------------
    ......2854495385411919762116571938898990272765503973
    ....z

    ....1
    + ----------------------------------------------------------------
    ......2854495385411919762116571938898990272765509231
    ....z

    ....1
    + ---------------------------------------------------------------...............(10b).
    ......2854495385411919762116571938898990272765509247
    ....z

(9) Prime 9-tuples
This again is an interlaced sequence where the first occurrence of a quadruplet is (5,23,41,59).

    9_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..100),k=9..9));

    ..................1.........1.......1........1......1......1......1.......1......1.......1
    9_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + ---+ ----...........(11).
    ....................5........23......41.....43.....53.....59....61.....71....79.....89
    ..................z.........z........z........z......z.......z........z.......z.......z.......z

(10) Prime 10-tuples
The first occurrence is a triplet given by (3,23,,43).

    10_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..100),k=10..10));

    ...................1.........1.......1
    10_tuple := ---- + ---- + ---- ..................................(12).
    .....................3........23......43
    ...................z.........z........z

(11) Prime 11-tuples
This tuple shows interlaced sequence formations. The first occurrence of a twin is (7,29).
    11_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=11..11));

    ...................1.........1.......1........1.......1......1.......1.......1
    11_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- ............................(13).
    .....................7........19......29......31.....37.....41....53.....59
    ...................z.........z........z.........z.......z.......z........z.......z

(12) Prime 12-tuples
This tuple shows interlaced sequence formations. The first occurrence of a quadruplet is (59,83,107,131).
    12_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..200),k=12..12));

    ...................1.........1........1........1.......1........1.........1........1........1.........1
    12_tuple := ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ----...........(14).
    ....................59........79......83.....103....107....127......131....151...349.....373
    ...................z..........z.........z........z.......z.........z..........z........z........z..........z

(13) Prime 13-tuples
This tuple shows interlaced sequence formations. The first occurrence of a twinis (3,29).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=13..13));

    ...................1........1........1........1........1........1.........1........1
    13_tuple := ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- ...........(15).
    ......................3.........5......11......17......29......31.......37.......43
    ...................z.........z.........z........z........z.........z..........z........z

(14) Prime 14-tuples
The first occurrence of a triplet is (3,31,59).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..20),k=14..14));

    ..................1.........1........1
    14_tuple := ---- + ---- + ---- ...........................................(16).
    .....................3.........31......59
    ..................z..........z.........z

(15) Prime 15-tuples
The first occurrence of a sextuplet (7,37,67,97,127,157).
    15_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..5)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..5),i=1..50),k=15..15));

    ...................1........1........1........1........1........1
    15_tuple := ---- + ---- + ---- + ---- + ---- + ---- ....................(17).
    ......................7........37......67......97.....127....157
    ...................z.........z.........z........z........z.........z

(16) Prime 16-tuples
This tuple shows interlaced sequence formations. The first occurrence of a twin is (5,37).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..50),k=16..16));

    ..................1.........1........1........1.........1........1..........1.......1
    16_tuple := ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- ..............(18).
    .....................5.........11......29......37.......41......43......47.......61
    ..................z..........z.........z.........z........z.........z..........z........z

(17) Prime 17-tuples
The first occurrence of a triplet is (3,37,71).
    17_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..50),k=17..17));

    ...................1........1........1
    17_tuple := ---- + ---- + ---- ...........(19).
    ......................3........37......71
    ...................z.........z.........z

(18) Prime 18-tuples
This tuple shows interlaced sequence formations. The first occurrence of a twin is (3,29).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..50),k=18..18));

    ...................1........1........1........1
    18_tuple := ---- + ---- + ---- + ---- ...........(20).
    .....................31........67....103....139
    ...................z.........z........z.........z

(19) Prime 19-tuples
This tuple shows interlaced sequence formations. The first occurrence of a twinis (3,29).
    19_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..50),k=19..19));

    ..................1.........1........1
    19_tuple := ---- + ---- + ---- ...........(21).
    .....................3.........41......79
    ..................z..........z.........z

(20) Prime 20-tuples
The first occurrence of a twinis (3,43,83).
    20_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..50),k=20..20));

    ..................1.........1........1
    20_tuple := ---- + ---- + ---- ...........(22).
    .....................3.........43......83
    ..................z..........z.........z

(21) Prime 21-tuples
The first occurrence of a quintuplet is (5,47,89,131,173).
    21_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..4)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..4),i=1..200),k=21..21));

    ...................1.........1.......1........1..........1
    21_tuple := ---- + ---- + ---- + ----- + ----- ...........(23).
    .....................5........47......89.......131.....173
    ...................z.........z........z.........z..........z

(22) Prime 22-tuples
The sequence shows interlaced formations. The first occurrence of a twin is (3,47).
    22_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=22..22));

    ...................1.........1.......1........1.......1.......1......1.......1
    22_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- ..................(24).
    .....................3.........17......23......29....47.....61....67.....73
    ...................z.........z........z.........z.......z........z.......z.......z

(23) Prime 23-tuples
The first occurrence of a twin is (7,53).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..30),k=23..23));

    ....................1.........1.......1.......1.......1.......1......1
    23_tuple := ---- + ---- + ---- + --- + --- + --- + ----...........(25).
    ......................7........13......37.....43......53.....59....83
    ....................z.........z........z........z.......z.......z.......z

(24) Prime 24-tuples
The first occurrence of a quintuplet is (5,53,101,149,197).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..4)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..4),i=1..30),k=24..24));

    ...................1.........2.......1........1........1
    24_tuple := ---- + ---- + ---- + --- + ----- ...........(26).
    .....................5........53.....101.....149....197
    ...................z.........z........z.........z........z

(25) Prime 25-tuples
The first occurrence of a triplet is (3,53,103).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..30),k=25..25));

    ...................1.........1.......1
    25_tuple := ---- + ---- + ---- ...........(27).
    .....................3.........53.....103
    ...................z.........z........z

(26) Prime 26-tuples
The first occurrence of a twin is (7,59).
    26_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..30),k=26..26));

    ...................1.........2.......1........1.......1.......1......1.......1
    26_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- ...........(28).
    .....................7........19......31......37.....59.....71....83.....89
    ...................z.........z........z.........z.......z.......z.......z.......z

(27) Prime 27-tuples
The first occurrence of a quadruplet is (5,59,113,157).
    27_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..100),k=27..27));

    ...................1.........2.......1........1.......1.......1......1.......1......1.......1.........1
    27_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + ---- + ----...........(29).
    .....................5........19......29......59....73.....83....113...127...137...167.....181
    ...................z.........z........z.........z.......z.......z.......z.......z........z.......z.........z

(28) Prime 28-tuples
The first occurrence of a twin is {3,59).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=28..28));

    ...................1.........1.......1........1.......1.......1......1.......1......1
    28_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- ...........(30).
    .....................3.........5......11.......17.....23....59.....61.....67.....73 ...................z.........z........z.........z.......z.......z.......z.......z........z

(29) Prime 29-tuples
The first occurrence of a twin is {3,61).
    29_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=29..29));

    ...................1.........1.......1........1.......1.......1
    29_tuple := ---- + ---- + ---- + --- + --- + --- ...........(31).
    .....................3........13......31......61.....71.....89 ...................z.........z........z.........z.......z.......z

(30) Prime 30-tuples
The first occurrence of a quintuplet is {11,71,131,191,251).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..5)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..5),i=1..300),k=30..30));

    ...................1.........1.......1.......1.......1.......1......1.......1.......1........1........1
    28_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + ---- + ---- ......(32).
    ....................11........53......71....113...131...173...191..233...251...293.....353 ...................z.........z........z........z.......z........z.......z.......z........z........z........z

(31) Prime 31-tuples
The first occurrence of a twin is {5,67).
    31_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=31..31));

    ...................1..........1.......1........1.......1.......1
    231_tuple := ---- + ---- + ---- + --- + --- + --- ...........(33).
    .....................5..........11......17......67.....73.....79 ...................z..........z........z.........z.......z.......z

(32) Prime 32-tuples
The first occurrence of a triplet is {3,67,131).
    32_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..20),k=32..32));

    ...................1.........1.......1
    32_tuple := ---- + ---- + ---- ...........(34).
    .....................3........67.....131
    ...................z.........z........z

(33) Prime 33-tuples
The first occurrence of a twin is {3,59).
    33_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..20),k=33..33));

    ...................1.........1........1........1
    33_tuple := ---- + ---- + ---- + --- ...........(35).
    ....................31........97......163.....229
    ...................z.........z.........z.........z

(34) Prime 34-tuples
The first occurrence of a triplet is {3,71,139).
    34_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..20),k=34..34));

    ...................1.........1.......1
    34_tuple := ---- + ---- + ---- ..........................(36).
    .....................3........71......139
    ...................z.........z........z

(35) Prime 35-tuples
The first occurrence of a twin is {3,73).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=35..35));

    ...................1.........1.......1........1.......1.......1......1.......1......1........1
    28_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + -----...........(37).
    .....................3........13......19......31.....37.....73.....83.....89....101....107 ...................z.........z........z.........z.......z.......z.......z.......z........z.........z

(36) Prime 36-tuples
The first occurrence of a quadruplet is {7,79,151,223)).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..20),k=36..36));

    ...................1.........1.......1........1
    28_tuple := ---- + ---- + ---- + ---...........(38).
    .....................7........79.....151.....223
    ...................z.........z........z.........z

(37) Prime 37-tuples
The first occurrence of a twin is {5,79).
    37_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=37..37));

    ...................1.........1.......1........1.......1.......1
    37_tuple := ---- + ---- + ---- + --- + --- + --- ...........(39).
    .....................5........23......29......79.....97....103 ...................z.........z........z.........z.......z.......z

(38) Prime 38-tuples
The first occurrence of a twin is {3,79).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=38..38));

    ...................1.........1.......1........1.......1.......1......1.......1......1........1
    28_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- + --- + -----...........(40).
    .....................3.........7......13......31.....37.....79.....83.....89....107....113 ...................z.........z........z.........z.......z.......z.......z.......z........z.......z

(39) Prime 39-tuples
The first occurrence of a twin is {23,101,179,257).
    39_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..20),k=39..39));

    ...................1.........1.......1..........1
    39_tuple := ---- + ---- + ---- + ----- ...........(41).
    ....................23.......101.....179.....257
    ...................z.........z........z..........z

(40) Prime 40-tuples
The first occurrence of a twin is {3,83,163).
    40_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..20),k=40..40));

    ...................1.........1.......1
    40_tuple := ---- + ---- + ---- ............................(42).
    .....................3........83......163
    ...................z.........z........z

(41) Prime 41-tuples
The first occurrence of a twin is {3,59).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=41..41));

    ...................1.........1.......1........1.......1.......1
    41_tuple := ---- + ---- + ---- + --- + --- + --- ...........(43).
    .....................7.........19......31......89...101....113 ...................z.........z........z.........z.......z.......z

(42) Prime 42-tuples
The first occurrence of a twin is {3,59).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..3)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..3),i=1..20),k=42..42));

    ...................1.........1.......1........1........1........1........1.........1
    42_tuple := ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- ...............(44).
    .....................5.........29......89.....113....173....197....257....281 ...................z.........z........z.........z........z.........z........z.........z

(43) Prime 43-tuples
The first occurrence of a twin is {3,59).
    43_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=43..43));

    ...................1.........1.......1........1.......1.......1......1.......1
    43_tuple := ---- + ---- + ---- + --- + --- + --- + --- + --- ...........(45).
    .....................3........11......17......23.....89.....97....103...109 ...................z.........z........z.........z.......z.......z.......z........z

(44) Prime 44-tuples
The first occurrence of a twin is {13,101).
    k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=44..44));

    ...................1.........1.......1...........1
    44_tuple := ---- + ---- + ------ + ------ ...........(46).
    .....................13........19.....101.....107
    ...................z.........z........z............z

(45) Prime 45-tuples
The first occurrence of a twin is {3,59).
    45_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..5)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..5),i=1..100),k=45..45));).

    ...................1.........1.......1........1.......1.........1
    45_tuple := ---- + ---- + ---- + ---- + ---- + ---- ...........(47)
    .....................13.......103...193....283...373.....463 ...................z.........z........z.........z.......z..........z

(46) Prime 46-tuples
The first occurrence of a twin is {5,97).
    46_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=46..46));

    ...................1.........1.......1........1.......1.......1
    46_tuple := ---- + ---- + ---- + --- + --- + --- ...........(48).
    .....................5........11......17......97....103....109 ...................z.........z........z.........z.......z.......z

(47) Prime 47-tuples
The first occurrence of a triplet is {3,97,191).
    47_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..20),k=47..47));

    ...................1.........1.......1
    47_tuple := ---- + ---- + ---- ...........(49).
    .....................3........97......191
    ...................z.........z........z

(48) Prime 48-tuples
The first occurrence of a quintuplet is {5,101,197,293,389).
    48_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..4)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..4),i=1..200),k=48..48));

    ...................1........1.......1........1.........1
    48_tuple := ---- + ---- + ---- + ---- + ---- .............................(50).
    .....................5.......101.....197...293.....389
    ...................z........z........z.........z.........z

(49) Prime 49-tuples
The first occurrence of a triplet is {5,101,199).
    49_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..2)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..2),i=1..200),k=49..49));

    ...................1.........1........1
    49_tuple := ---- + ---- + ---- ...............................(51).
    .....................5........101....199
    ...................z.........z.........z

(50) Prime 50-tuples
The first occurrence of a twin is {3,103).
    50_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..1)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..1),i=1..20),k=50..50));

    ...................1.........1.......1........1.......1.......1.......1.......1........1........1
    50_tuple := ---- + ---- + ---- + --- + --- + --- + ---- + --- + --- + ---...........(52).
    .....................3........7........13......31.....37....103....107....113....131....137
    ...................z.........z........z.........z.......z........z........z........z........z........z



4. Search Beyond 50-tuples

From the k-tuples listed in Table 1, it is noticed that long tuples occur only amongst those with the value of k divisible by 3 but the longest is found in k = 15. Subsequent search by the Maple program line given below confirms that long tuples occur with values of k ending either in zeroes or fives. The results are summarised in Table 2.

for k from 50 by 1 to 100 do print(k): sum(sum(1/z^(2*i-1+2*j*k),j=0..5)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..5),i =1..1000) od;.........................................(53).

Table 2 - First Occurrences of Sextuplet to Decutuplet for specific values of k.
The following values of k generate sextuplets:
    60 90 120 135 150 165 180 195 225 240 255 270 285 315 330 345 360 375 390 405 450 465 495 510 540 570 585 600 645 660 675 705 720 735 780 825 840 855 870 885 900 915 960 990...................(54).
75-tuples: A septupulet (7,157,307,457,607,757,907) is found.
75_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..6)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..6),i=1..2000),k=75..75));

.....................1......1.......1..........1.........1.........1.........1
75_tuple := ----- + ---- + ---- + ---- + ---- + ---- + ----...........(55).
.......................7.....157.....307.....457.....607.....757.....907
.....................z......z........z...........z..........z.........z.........z

105-tuples: A decupulet (199,409,519,829,1039,1249,1459,1669,1879,2089) is found.
k_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..9)*product((isprime(2*i-1+2*j*k)-false)/ (true-false),j=0..9),i=1..1000),k=105..105));

.....................1........1.........1.......1...........1.........1.........1..........1.........1...........1
105_tuple := ---- + ---- + ---- + ---- + ----- + ----- + ----- + ----- + ----- + -----.....(56).
......................199....409.....619.....829....1039...1249....1459....1669...1879...2089
....................z........z..........z.........z...........z..........z..........z..........z..........z...........z

630_tuples: A nonutuplet: (2063,3323,4583,5843,7103,8363,9623,10883,12143).
630_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..8)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..8),i=1..2000),k=630..630));

.....................1..........1.........1..........1..........1..........1...........1..........1...........1
630_tuple := ------ + ------ + ----- + ----- + ----- + ----- + ----- + ------ + ------....(57).
......................2063....3323.....4583....5843....7103....8363....9623...10883...12143
.....................z..........z..........z...........z...........z..........z...........z..........z.............z

945-tuple is an octuplet given by: (2693,4583,6473,8363,10253,12143,14033,15923)
945_tuple:=sort(sum(sum(sum(1/z^(2*i-1+2*j*k),j=0..7)*product((isprime(2*i-1+2*j*k)-false)/(true-false),j=0..7),i=1..2000),k=945..945));

    .....................1..........1.........1..........1..........1...........1............1............1
    945_tuple := ----- + ----- + ----- + ----- + ------ + ------ + ------ + ------..(58).
    ......................2693....4583....6473...8363...10253....12143....14033....15923
    ....................z..........z...........z..........z..........z............z............z.............z



5. Conclusions

The search of regular prime k-tuples is confined within the range for k = 1 to 1000. Only the first occurrence of each long tuple is reported and this always occurs right at the beginning of the sequence. The definition of regular k-tuple seems to give consistency to the family of tuples which include the classical twinprimes. The common property detected amongst these regular k-tuples is that the longest tuple always occur at the beginning of the sequence. Long tuples appear to be associated with values of k which are divisible by 15. This means that k values which are perfectly divisible by 15 will generate long k-tuples. There is no proof that these tuples are always finite but considering that the intervals between primes are nonlinear, it is unlikely that any sequence with regular interval would fit a subset from the global prime set.


6. References

(1) For further information please click: The Prime k-tuplet Hompage by Tony Forbes


2. Global Generating Function For Palindromic Products of Consecutive Integers - Huen Y.K. (Date Released 18/2/98, 21 Kbytes)

3. Sequence Algebra - A Tutorial Paper - Huen Y.K. (Date Released 2/2/98, 46 Kbytes)


Published Papers:

4. 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.

5. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

6. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol., 1997, VOL.28, NO.3, pp 333-344.

7. Huen Y.K.: The twin prime problem revisited, INT. J. Math. Educ. Sci. Technol.,1997, VOL.28, NO. 6, pp 825-834.

8. Huen Y.K.: Is Pie Periodic?, INT. J. Math. Educ. Sci. Technol.,199?,VOL.??,NO.?,???-???. (in the press).

9. Huen Y.K.: Final value theorem in number sequences., INT. J. Math. Educ. Sci. Technol.,199?,VOL.-??,NO.?,???-???. (accepted).

Papers posted in this website.

Comments: References from this point onward are not referred in the main paper. Most are provided for readers not familiar with sequence algebra. These papers can be easily hyperlinked whilst you surf into this URLsite.

10. A Simple Introduction To Sequence Algebra - by Huen Y.K. (date release: 15.3.97) (38 KBytes, 11*A4 pages).

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11. Lemmata, Corollaries, And Theorems In Sequence Order Analysis. - by Huen Y.K. (date released : 6.7.97) (38.3 KBytes, 12*A4s).

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12. 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 ).

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13. Generating Functions - Closed Forms vs Open Forms - by Huen Y.K. (date released : 1.10.97 ) (21 Kbytes).

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14. A Sequence Algebraist's View Of Lehmann's Primality Test - by Huen Y.K. (date released : 6.10.97 ) (26 Kbytes).

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