Test page will calculate
α = 0 to α = 50
λ = 0 to λ = 35.000

Note: On table 2,3,4 and 5 we only show λ = 0 to λ = 10

Click on a Contents item and be automatically transferred to the item itself via a hyperlink.

# Introduction

All famous Mathematicians, from Leonhard Euler to Terence Tao, have concluded that Prime Numbers sprout out randomly among the natural numbers and that, according to Euler "Mathematicians have tried in vain to discover some order in the sequence of Primes and we have reason to believe that it is a mystery into which the mind will never penetrate".

Yet, there is an Order in the Sequence of the Primes and the Prime Numbers exhibit stunning regularity, ie there are Laws governing their behaviour and the Primes obey these Laws with Military Precision and not "almost military precision" as Don Zagier suspected in a 1975 lecture of his https://www.academia.edu/7616341/Genesis_16_-_Statements_by_famous_Mathematicians_on_Prime_numbers

All Prime Numbers, bar 2 and 3, reside in the [ 6α + 5 ] or [ 6α + 7 ] one column Matrices, the Primes Residences,

Constantine Adraktas, an Alumnus of MIT, has discovered four (4) Matrix formulae, which are the basis for the Order in the Sequence of the Prime Numbers. The formulae identify the Composites cohabitants of the Primes in the Primes Residences [ 6α + 5 ] and [ 6α + 7 ] one column vectors.

These are the four (4) Constantine Adraktas Matrix formulae

Logic for finding the Composite cohabitants of the Primes in the Primes Residencies Matrix Equations Primes Residences
1 Multiply each element of [ 6α + 5 ] with each element of [ 6α + 5 ] leading to [ 6α + 5 ] * [ 6 * ( α + λ ) + 5 ]T and ending up in [ 6α + 7 ]
2 Multiply each element of [ 6α + 5 ] with each element of [ 6α + 7 ] leading to [ 6α + 5 ] * [ 6 * ( α + λ) + 7 ]T and ending up in [ 6α + 5 ]
3 Multiply each element of [ 6α + 7 ] with each element of [ 6α + 5 ] leading to [ 6α + 7 ] * [ 6 * ( α + λ) + 5 ]T and ending up in [ 6α + 5 ]
4 Multiply each element of [ 6α + 7 ] with each element of [ 6α + 7 ] leading to [ 6α + 7 ] * [ 6 * ( α + λ) + 7 ]T and ending up in [ 6α + 7 ]

His fellow Greek Eratosthenes did not develop formulae for his famous Sieve.

His good friend Alper Sümer Güc, an expert software engineer, has developed this site which relies on the above mentioned Matrix formulae. Four mysql tables covering the above set of Composite numbers, cohabitants of the Primes in [ 6α + 5 ] and [ 6α + 7 ], emerge.

Follow the Tables and enjoy the Military Precision based Order of Sequence of the Prime Numbers.

The α maximum of 50 of the above widow is for visual presentation reasons.

Alper Sümer Güc has, also, developed a site, namely https://www.isprimenumber.com/ , which is not based on any formulae but rather on a database of the first 5,000,000 Primes . This site tells you if a positive integer is a Prime or not, within the said database.

He has, also, developed https://www.isprimenumber.com/checker.php which based on the Constantine Adraktas Matrix formulae.

Time: 05:38:42.54040300
Memory usage: 235.02 KB

### 1. The Primes and their Composites cohabitants in the [ 6α + 5 ] and [ 6α + 7 ] Column Vector Residences

[ α ][ 6α + 5 ][ 6α + 7 ][ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 2. The α * λ Matrix for the 1st set of the Composites cohabitants of the Primes in the [ 6α + 7 ] Column Vector Residence

ending in
[ 6α + 7 ]
[ α ] [ 6α+5 ]
times
{ [ 0 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 1 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 2 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 3 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 4 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 5 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 6 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 7 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 8 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 9 * 6 ] + [ 6α+5 ] }
[ 6α+5 ]
times
{ [ 10 * 6 ] + [ 6α+5 ] }
70255585115145175205235265295325
131121187253319385451517583649715781
1922893914935956977999011003110512071309
2535296678059431081121913571495163317711909
3148411015118913631537171118852059223324072581
37512251435164518552065227524852695290531153325
43616811927217324192665291131573403364938954141
49722092491277330553337361939014183446547475029
55828093127344537634081439947175035535356715989
61934813835418945434897525156055959631366677021
671042254615500553955785617565656955734577358125
731150415467589363196745717175978023844988759301
79125929639168537315777782398701916396251008710549
8513688973877885838388819379987710375108731137111869
9114792184558989952310057105911112511659121931272713261
971590259595101651073511305118751244513015135851415514725
103161020110807114131201912625132311383714443150491565516261
109171144912091127331337514017146591530115943165851722717869
115181276913447141251480315481161591683717515181931887119549
121191416114875155891630317017177311844519159198732058721301
127201562516375171251787518625193752012520875216252237523125
133211716117947187331951920305210912187722663234492423525021
139221876919591204132123522057228792370124523253452616726989
145232044921307221652302323881247392559726455273132817129029
151242220123095239892488325777266712756528459293533024731141
157252402524955258852681527745286752960530535314653239533325
163262592126887278532881929785307513171732683336493461535581
169272788928891298933089531897328993390134903359053690737909
175282992930967320053304334081351193615737195382333927140309
181293204133115341893526336337374113848539559406334170742781
187303422535335364453755538665397754088541995431054421545325
193313648137627387733991941065422114335744503456494679547941
199323880939991411734235543537447194590147083482654944750629
205334120942427436454486346081472994851749735509535217153389
211344368144935461894744348697499515120552459537135496756221
217354622547515488055009551385526755396555255565455783559125
223364884150167514935281954145554715679758123594496077562101
229375152952891542535561556977583395970161063624256378765149
235385428955687570855848359881612796267764075654736687168269
241395712158555599896142362857642916572567159685937002771461
247406002561495629656443565905673756884570315717857325574725
253416300164507660136751969025705317203773543750497655578061
259426604967591691337067572217737597530176843783857992781469
265436916970747723257390375481770597863780215817938337184949
271447236173975755897720378817804318204583659852738688788501
277457562577275789258057582225838758552587175888259047592125
283467896180647823338401985705873918907790763924499413595821
289478236984091858138753589257909799270194423961459786799589
29548858498760789365911239288194639963979815599913101671103429
30149894019119592989947839657798371100165101959103753105547107341
3075093025948559668598515100345102175104005105835107665109495111325

### 2a. The Column Vector for the 1st set of the Composites cohabitants of the Primes in the [ 6α + 7 ] Column Vector Residence

1st set of Composites in
[ α ][ 6α + 7 ]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

### 3. The α * λ Matrix for the 1st set of the Composites cohabitants of the Primes in the [ 6α + 5 ] column vector Residence

ending in
[ 6α + 5 ]
[ α ] [ 6α+5 ]
times
{ [ 0 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 1 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 2 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 3 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 4 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 5 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 6 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 7 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 8 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 9 * 6 ] + [ 6α+7 ] }
[ 6α+5 ]
times
{ [ 10 * 6 ] + [ 6α+7 ] }
50356595125155185215245275305335
111143209275341407473539605671737803
1723234255276297318339351037113912411343
2335757138519891127126514031541167918171955
2948991073124714211595176919432117229124652639
35512951505171519252135234525552765297531853395
41617632009225525012747299332393485373139774223
47723032585286731493431371339954277455948415123
53829153233355138694187450548235141545957776095
59935993953430746615015536957236077643167857139
651043554745513555255915630566957085747578658255
711151835609603564616887731377398165859190179443
77126083654570077469793183938855931797791024110703
83137055755380518549904795451004310541110391153712035
8914809986339167970110235107691130311837123711290513439
951592159785103551092511495120651263513205137751434514915
101161040311009116151222112827134331403914645152511585716463
107171166312305129471358914231148731551516157167991744118083
113181299513673143511502915707163851706317741184191909719775
119191439915113158271654117255179691868319397201112082521539
125201587516625173751812518875196252037521125218752262523375
131211742318209189951978120567213532213922925237112449725283
137221904319865206872150922331231532397524797256192644127263
143232073521593224512330924167250252588326741275992845729315
149242249923393242872518126075269692786328757296513054531439
155252433525265261952712528055289852991530845317753270533635
161262624327209281752914130107310733203933005339713493735903
167272822329225302273122932231332333423535237362393724138243
173283027531313323513338934427354653650337541385793961740655
179293239933473345473562136695377693884339917409914206543139
185303459535705368153792539035401454125542365434754458545695
191313686338009391554030141447425934373944885460314717748323
197323920340385415674274943931451134629547477486594984151023
203334161542833440514526946487477054892350141513595257753795
209344409945353466074786149115503695162352877541315538556639
215354665547945492355052551815531055439555685569755826559555
221364928350609519355326154587559135723958565598916121762543
227375198353345547075606957431587936015561517628796424165603
233385475556153575515894960347617456314364541659396733768735
239395759959033604676190163335647696620367637690717050571939
245406051561985634556492566395678656933570805722757374575215
251416350365009665156802169527710337253974045755517705778563
257426656368105696477118972731742737581577357788998044181983
263436969571273728517442976007775857916380741823198389785475
269447289974513761277774179355809698258384197858118742589039
275457617577825794758112582775844258607587725893759102592675
281467952381209828958458186267879538963991325930119469796383
2874782943846658638788109898319155393275949979671998441100163
293488643588193899519170993467952259698398741100499102257104015
29949899999179393587953819717598969100763102557104351106145107939
3055093635954659729599125100955102785104615106445108275110105111935

### 3a. The Column Vector for the 1st set of the Composites cohabitants of the Primes in the [ 6α + 5 ] Column Vector Residence

1st set of Composites in
[ α ][ 6α + 5 ]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

### 4. The α * λ Matrix for the 2nd set of the Composites cohabitants of the Primes in the [ 6α + 5 ] column vector Residence

ending in
[ 6α + 5 ]
[ α ] [ 6α+7 ]
times
{ [ 0 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 1 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 2 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 3 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 4 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 5 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 6 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 7 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 8 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 9 * 6 ] + [ 6α+5 ] }
[ 6α+7 ]
times
{ [ 10 * 6 ] + [ 6α+5 ] }
503577119161203245287329371413455
111143221299377455533611689767845923
17232343755166577989310071121123513491463
23357572587510251175132514751625177519252075
2948991085127114571643182920152201238725732759
35512951517173919612183240526272849307132933515
41617632021227925372795305333113569382740854343
47723032597289131853479377340674361465549495243
53829153245357539054235456548955225555558856215
59935993965433146975063542957956161652768937259
651043554757515955615963636567677169757179738375
711151835621605964976935737378118249868791259563
77126083655770317505797984538927940198751034910823
83137055756580758585909596051011510625111351164512155
8914809986459191973710283108291137511921124671301313559
951592159797103791096111543121251270713289138711445315035
101161040311021116391225712875134931411114729153471596516583
107171166312317129711362514279149331558716241168951754918203
113181299513685143751506515755164451713517825185151920519895
119191439915125158511657717303180291875519481202072093321659
125201587516637173991816118923196852044721209219712273323495
131211742318221190191981720615214132221123009238072460525403
137221904319877207112154522379232132404724881257152654927383
143232073521605224752334524215250852595526825276952856529435
149242249923405243112521726123270292793528841297473065331559
155252433525277262192716128103290452998730929318713281333755
161262624327221281992917730155311333211133089340673504536023
167272822329237302513126532279332933430735321363353734938363
173283027531325323753342534475355253657537625386753972540775
179293239933485345713565736743378293891540001410874217343259
185303459535717368393796139083402054132742449435714469345815
191313686338021391794033741495426534381144969461274728548443
197323920340397415914278543979451734636747561487554994951143
203334161542845440754530546535477654899550225514555268553915
209344409945365466314789749163504295169552961542275549356759
215354665547957492595056151863531655446755769570715837359675
221364928350621519595329754635559735731158649599876132562663
227375198353357547315610557479588536022761601629756434965723
233385475556165575755898560395618056321564625660356744568855
239395759959045604916193763383648296627567721691677061372059
245406051561997634796496166443679256940770889723717385375335
251416350365021665396805769575710937261174129756477716578683
257426656368117696717122572779743337588777441789958054982103
263436969571285728757446576055776457923580825824158400585595
269447289974525761517777779403810298265584281859078753389159
275457617577837794998116182823844858614787809894719113392795
281467952381221829198461786315880138971191409931079480596503
2874782943846778641188145898799161393347950819681598549100283
293488643588205899759174593515952859705598825100595102365104135
29949899999180593611954179722399029100835102641104447106253108059
3055093635954779731999161101003102845104687106529108371110213112055

### 4a. The Column Vector for the 1st set of the Composites cohabitants of the Primes in the [ 6α + 5 ] Column Vector Residence

2nd set of Composites in
[ α ][ 6α + 5 ]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

### 5. The α * λ Matrix for the 2nd set of the Composites cohabitants of the Primes in the [ 6α + 7 ] column vector Residence

ending in
[ 6α + 7 ]
[ α ] [ 6α+7 ]
times
{ [0*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [1*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [2*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [3*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [4*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [5*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [6*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [7*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [8*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [9*6] + [ 6α+7 ] }
[ 6α+7 ]
times
{ [10*6] + [ 6α+7 ] }
704991133175217259301343385427469
131169247325403481559637715793871949
19236147558970381793110451159127313871501
25362577592510751225137515251675182519752125
3149611147133315191705189120772263244926352821
37513691591181320352257247927012923314533673589
43618492107236526232881313933973655391341714429
49724012695298932833577387141654459475350475341
55830253355368540154345467550055335566559956325
61937214087445348195185555159176283664970157381
671044894891529356956097649969017303770581078509
731153295767620566437081751979578395883392719709
791262416715718976638137861190859559100331050710981
85137225773582458755926597751028510795113051181512325
9114828188279373991910465110111155712103126491319513741
971594099991105731115511737123191290113483140651464715229
103161060911227118451246313081136991431714935155531617116789
109171188112535131891384314497151511580516459171131776718421
115181322513915146051529515985166751736518055187451943520125
121191464115367160931681917545182711899719723204492117521901
127201612916891176531841519177199392070121463222252298723749
133211768918487192852008320881216792247723275240732487125669
139221932120155209892182322657234912432525159259932682727661
145232102521895227652363524505253752624527115279852885529725
151242280123707246132551926425273312823729143300493095531861
157252464925591265332747528417293593030131243321853312734069
163262656927547285252950330481314593243733415343933537136349
169272856129575305893160332617336313464535659366733768738701
175283062531675327253377534825358753692537975390254007541125
181293276133847349333601937105381913927740363414494253543621
187303496936091372133833539457405794170142823439454506746189
193313724938407395654072341881430394419745355465134767148829
199323960140795419894318344377455714676547959491535034751541
205334202543255444854571546945481754940550635518655309554325
211344452145787470534831949585508515211753383546495591557181
217354708948391496935099552297535995490156203575055880760109
223364972951067524055374355081564195775759095604336177163109
229375244153815551895656357937593116068562059634336480766181
235385522556635580455945560865622756368565095665056791569325
241395808159527609736241963865653116675768203696497109572541
247406100962491639736545566937684196990171383728657434775829
253416400965527670456856370081715997311774635761537767179189
259426708168635701897174373297748517640577959795138106782621
265437022571815734057499576585781757976581355829458453586125
271447344175067766937831979945815718319784823864498807589701
277457672978391800538171583377850398670188363900259168793349
283468008981787834858518386881885799027791975936739537197069
2894783521852558698988723904579219193925956599739399127100861
295488702588795905659233594105958759764599415101185102955104725
30149906019240794213960199782599631101437103243105049106855108661
3075094249960919793399775101617103459105301107143108985110827112669

### 5a. The Column Vector for the 2nd set of the Composites cohabitants of the Primes in the [ 6α + 7 ] Column Vector Residence

2nd set of Composites in
[ α ][ 6α + 7 ]
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

### 6. The Column Vectors for the 1st and 2nd sets of the Composites cohabitants of the Primes in the [ 6α + 5 ] and [ 6α + 7 ] Column Vector Residences

1st set of Composites in 2nd set of Composites in 1st set of Composites in 2nd set of Composites in
[ α ] [ 6α + 5 ] [ 6α + 5 ] [ 6α + 7 ] [ 6α + 7 ] [ α ]
055770
1111113131
2171719192
3232325253
4292931314
5353537375
6414143436
7474749497
8535355558
9595961619
106565676710
117171737311
127777797912
138383858513
148989919114
159595979715
1610110110310316
1710710710910917
1811311311511518
1911911912112119
2012512512712720
2113113113313321
2213713713913922
2314314314514523
2414914915115124
2515515515715725
2616116116316326
2716716716916927
2817317317517528
2917917918118129
3018518518718730
3119119119319331
3219719719919932
3320320320520533
3420920921121134
3521521521721735
3622122122322336
3722722722922937
3823323323523538
3923923924124139
4024524524724740
4125125125325341
4225725725925942
4326326326526543
4426926927127144
4527527527727745
4628128128328346
4728728728928947
4829329329529548
4929929930130149
5030530530730750

### 7. The Column Vectors for the Composites cohabitants of the Primes in the [ 6α + 5 ] and [ 6α + 7 ] Column Vector Residences

Composites in Composites in
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 8. Primes in the [ 6α + 5 ] Column Vector

Primes + Composites in minus Composites in equals Primes in
[ α ] [ 6α + 5 ] [ 6α + 5 ] [ 6α + 5 ] [ α ]
05550
11111111
21717172
32323233
42929294
53535355
64141416
74747477
85353538
95959599
1065656510
1171717111
1277777712
1383838313
1489898914
1595959515
1610110110116
1710710710717
1811311311318
1911911911919
2012512512520
2113113113121
2213713713722
2314314314323
2414914914924
2515515515525
2616116116126
2716716716727
2817317317328
2917917917929
3018518518530
3119119119131
3219719719732
3320320320333
3420920920934
3521521521535
3622122122136
3722722722737
3823323323338
3923923923939
4024524524540
4125125125141
4225725725742
4326326326343
4426926926944
4527527527545
4628128128146
4728728728747
4829329329348
4929929929949
5030530530550

### 9. Primes in the [ 6α + 7 ] Column Vector

Primes + Composites in minus Composites in equals Primes in
[ α ] [ 6α + 7 ] [ 6α + 7 ] [ 6α + 7 ] [ α ]
07770
11313131
21919192
32525253
43131314
53737375
64343436
74949497
85555558
96161619
1067676710
1173737311
1279797912
1385858513
1491919114
1597979715
1610310310316
1710910910917
1811511511518
1912112112119
2012712712720
2113313313321
2213913913922
2314514514523
2415115115124
2515715715725
2616316316326
2716916916927
2817517517528
2918118118129
3018718718730
3119319319331
3219919919932
3320520520533
3421121121134
3521721721735
3622322322336
3722922922937
3823523523538
3924124124139
4024724724740
4125325325341
4225925925942
4326526526543
4427127127144
4527727727745
4628328328346
4728928928947
4829529529548
4930130130149
5030730730750

### 10. Primes in the [ 6α + 5 ] and [ 6α + 7 ] Column Vector

Primes in Primes in
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 11. Single Primes in the [ 6α + 5 ] + [ 6α + 7 ] Two Column Vector

Single Primes in Single Primes in
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 12. Twin Primes in the [ 6α + 5 ] + [ 6α + 7 ] Two Column Vector

Members of Twin Primes
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 13. Single Composites in the [ 6α + 5 ] + [ 6α + 7 ] Two Column Vector

Single Composites in Single Composites in
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 14. Twin Composites in the [ 6α + 5 ] + [ 6α + 7 ] Two Column Vector

Members of Twin Composites
[ α ] [ 6α + 5 ] [ 6α + 7 ] [ α ]
0570
111131
217192
323253
429314
535375
641436
747497
853558
959619
10656710
11717311
12777912
13838513
14899114
15959715
1610110316
1710710917
1811311518
1911912119
2012512720
2113113321
2213713922
2314314523
2414915124
2515515725
2616116326
2716716927
2817317528
2917918129
3018518730
3119119331
3219719932
3320320533
3420921134
3521521735
3622122336
3722722937
3823323538
3923924139
4024524740
4125125341
4225725942
4326326543
4426927144
4527527745
4628128346
4728728947
4829329548
4929930149
5030530750

### 15. The ∆ = 2 pairing structure of Primes and Composites in the Primes Residences

 [ 6α + 5 ] [ 6α + 7 ] Single Composites Single Primes Single Primes Single Composites Member of Twin Primes Member of Twin Primes Member of Twin Composites Member of Twin Composites

### 15a. The kaleidoscope of Primes and Composites in the Primes Residences - See 15 for colour code

Composites and PrimesComposites and Primes
[ α ]in [ 6α + 5 ]in [ 6α + 7 ]
057
11113
21719
32325
42931
53537
64143
74749
85355
95961
106567
117173
127779
138385
148991
159597
16101103
17107109
18113115
19119121
20125127
21131133
22137139
23143145
24149151
25155157
26161163
27167169
28173175
29179181
30185187
31191193
32197199
33203205
34209211
35215217
36221223
37227229
38233235
39239241
40245247
41251253
42257259
43263265
44269271
45275277
46281283
47287289
48293295
49299301
50305307

Calculated in 0.0135 seconds.

Time: 05:38:42.55327400