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Brilliant numbers

Brilliant numbers, as defined by Peter Wallrodt, are numbers with two prime factors of the same length (in decimal notation). These numbers are generally used for cryptographic purposes, and for testing the performance of prime factoring programs.

The challenge is to continue the table shown below. This table contains the least brilliant number with an even number of digits and the greatest brilliant number with an odd number of digits.

It is possible to define n-brilliant numbers as the product of n prime numbers of the same length.

Table of contents

Maximal and minimal 2-brilliant numbers

The next table contains the maximal and minimal numbers with given number of figures whose two prime factors have the same length when expressed in decimal notation.

Maximal and minimal 2-brilliant numbers
DigitsNumberPrime factorsDiscoverer
193Jim Fougeron
3
2102
5
398923
43
4100317
59
599973257
389
6100013103
971
7107 - 1891583
6317
8107 + 432089
4787
9109 - 13725303
39521
10109 + 8126881
37201
111011 - 357231571
431833
121011 + 147281683
355009
131013 - 771380163
7245521
141013 + 731857929
5382337
151015 - 26118874013Richard Heylen
52982903
161015 + 314902357
67103479
171017 - 297150425719Richard Heylen
664779937
181017 + 831158267567
631841393
191019 - 5331809444697Richard Heylen
5526557411
201019 + 491172005397
8532383917
211021 - 35321942092327Richard Heylen
45574505161
221021 + 98710119972739
98814495433
231023 - 587198168943561Richard Heylen
504619937933
241023 + 691253850475563
393932687257
251025 - 1132701224339427Richard Heylen
3702024986981
261025 + 1831634713216901
6117280937483
271027 - 28116816039284931Richard Heylen
59467035195149
281027 + 415310594044136579
94392659413907
291029 - 93273350346504617Richard Heylen
365830888011371
301029 + 279262922036885471
380340884258249
311031 - 3891856935855186693Richard Heylen
5385215634707327
321031 + 6671178160270652879
8487809552819573
331033 - 108327586826038690693Richard Heylen
36249186426792769
341033 + 70928567159477900867
35005230421091927
351035 - 807260929504743206393Richard Heylen
383245275762949601
361035 + 277312315473725119131Jim Fougeron
320189066546263567
371037 - 7071124073940762336703Richard Heylen
8896211928210070531
381037 + 16872918984460772347259Jim Fougeron
3425849001386617493
391039 - 107122168728539128208483Richard Heylen
45108586098430581563
401039 + 99718288834983913311471Jim Fougeron
54678168449744921707
411041 - 3699127062416457894753317Richard Heylen
787014782086546036553
421041 + 1207228728453883196320979Jim Fougeron
437199650075309509133
431043 - 272551499368987998869591Richard Heylen
3919264147796477703203
441043 + 912094162205298508378983Jim Fougeron
4775179293513497904877
451045 - 1715380127434630196194863Richard Heylen
65018967121714506226241
461045 + 141120454925622447906868371Jim Fougeron
48887980257555528444241
471047 - 1677279170318528058249521813Richard Heylen
358204269448327527799271
481047 + 393205522560092430172482139Jim Fougeron
486564589089522586263787
491049 - 4711943643268293290666134453Richard Heylen
5144977045495071915161893
501049 + 9512647141521862657404330791Jim Fougeron
3777659757670800343659761
511051 - 57929121661308190630398411623Richard Heylen
34338700303431672610407227
521051 + 979313953377225420926914498491Jim Fougeron
71667237532871425027689523
531053 - 167185862635189918295875711699Richard Heylen
538031756075221497635484067
541053 + 2217140830196310821661891328463Jim Fougeron
710074988316378627095268359
551055 - 2671049230952180343004616980607Richard Heylen
9530790127015990538677972619
561055 + 62291460201993355564860764167713Jim Fougeron
6848367585788496545968810933
571057 - 834924470570298372015475229439271Jim Fougeron
40865414569701641261618151781
581057 + 231721827649121989366053485370671Jim Fougeron
45813454046803015025118660227
591059 - 5973100989110031147659340436549999Jim Fougeron
990205775347038981060013155973
601059 + 213213916881789829278910570173437
467471286806848547076331371449
611061 - 691055217882882115232154201320731Bouk de Water
9476715815966853211400570143201
621061 + 3991960582085021883125774065415291Bouk de Water
5100526051113225665789755534589
631063 - 914499216344898896752959323769763Bouk de Water
68969244696581083712684424134557
641063 + 1922350568628980377780095228766733Jim Fougeron
44741590990368441481914582608543
651065 - 687158997176633105910786856570509671Bouk de Water
628941985748307323903185090439303
661065 + 2317286595216834133465028509515988163Jim Fougeron
348924176420832745825986935358959
671067 - 8091250511065160951547577917550796961Jim Fougeron
7996730519703889103364386584269431
681067 + 6091678092783518480469813357050416541Jim Fougeron
5959146060465656117945029500096149
691069 - 124110679904962845456862138386056396401Phil Carmody
93633792012093812114892571488571159
701069 + 260724426597353643084809987417412256391Jim Fougeron
40938980797129151552564164844215577
711071 - 7619231008270216639494612708088690827091Jim Fougeron
432884934838999601577138575168198191
721071 + 11901162435955060234923989750540378209139Jim Fougeron
615627248061660606526348016196129359
731073 - 82972279546384325209074657090630001002097Richard Heylen
4386837692254372909233641841447115399
741073 + 105631093872132886178625296967462523488383Jim Fougeron
9141836325617901427873699501869702461
751075 - 420911603482293180564213674824122706652977Richard Heylen
86181025207209215754156369211739413183
761075 + 547311171602016259629892335959089815716691Jim Fougeron
(2 May 2002)
89512676744531089226201976823759569403
771077 - 4851266574540546930479652420365114117132441Richard Heylen
(10 May 2002)
375129597128181069020562559344084771989
781077 + 3185554311496620532371770351611143673123Jim Fougeron
538925768921415130739706333069119686561
791079 - 56671729776145369695994282075176946198769503Richard Heylen
5781094869858291625387593074971223276611
801079 + 59231333322076518899001350381760807974795003Richard Heylen
(16 May 2002)
7500063320115780212377802894180923803641
811081 - 498928090949127071263736011460619683106498869Michael Bell
35598654765149940348772510523335003778519
821081 + 1752716185521526191837352841378940110747187933Richard Heylen
61783613112606452271787097523839696139619
831083 - 14333199932327974298861069048863455316009796077Richard Heylen
(27 Jun 2002)
500169237327416700002311820952289282634671
841083 + 8569132087033835831918079103641067858862243541Richard Heylen
(27 Jun 2002)
757076581220589828720159898116433821618709
851085 - 70071693200872034169587540836261917796700786223Richard Heylen
(8 Jul 2002)
5905973806868081574548312122783044390802991
861085 + 167013051275553674272224126253462719544495568863Richard Heylen
(27 Sep 2002)
3277317903313662333191415700136829929609827
871087 - 611316720583183841033646029730324154154355596829Richard Heylen
(11 Oct 2002)
59806526423457026799428621903555375545514603
881087 + 1198925548073574973342022878007235853527655387529Reto Keiser
(25 Jul 2002)
39141894478478048759397262019204591690357741
891089 - 14547171825942024805992446123879964620028962720089Richard Heylen
(24 Nov 2002)
581984296559615555233094293492402617340927477
901089 + 9757125449117210655591881673183684137233747818249Richard Heylen
(24 Nov 2002)
797135940240048534567034950286701267149637493
911091 - 355792978190131233156687766899939838042396357404141David Cleaver
(5 Sep 2002)
3357743985223460380218440581090317089334569081
921091 + 64893030426176188825904708253480783779093519047793David Cleaver
(11 Nov 2002)
3299865899579960543508682793462789222873063273
931093 - 1364313086519402050960470783685012210223925173521497David Cleaver
(9 Dec 2002)
76414512467178771426083642931523688230774568381
941093 + 348929411667593005444701023975598270078264073129867Jim Fougeron
(9 Dec 2002)
34000112262856379670017139669568496330607787267
951095 - 3681111097491776794870937630002243840967616045495643Richard Heylen
(30 Jan 2003)
900110330131568068227760865309719427458876830733
961095 + 2899136300932834042408394024592335302109536892634161Jim Fougeron
(12 Dec 2002)
733670694108588540814651977458574150528292043459
971097 - 191631507302631098002948390959131169706059408828799543David Cleaver
(20 Mar 2003)
6634367773056592241515494527842465247176344374659
981097 + 125771248259542695921544034940904194773338751647883781Jim Fougeron
(3 Jan 2003)
8011154457832187771253603613895408568341998334317
991099 - 278331084801612134411476333699496627802172660847517087David Cleaver
(13 Dec 2002)
32170062156987845422435581553308407138632546972991
1001099 + 401123156485057650669192768497163340558192130875875893David Cleaver
(20 Dec 2002)
43184446927518910400647566770334520938655251359327
10110101 - 56993117824543400831857004264281890859168944172987388481David Cleaver
(4 Mar 2003)
848719605556256173460898892513108240280912733932447
10210101 + 3231227019196535950614963176497989255069045234236491673Richard Heylen
(13 Jul 2003)
440491383662192037893868941350132265860839934720247
10310103 - 24172388684044013249161833428010656139781947511337567461David Cleaver
(20 Jul 2003)
4186405491786561972002257851552363665292291130223203
10410103 + 173892178850329744203343440104133177891981618919402435883David Cleaver
(17 Aug 2003)
4589576375892693047888073495994066512714871682387783
10510105 - 3017920072845108389786097528973616362100626048357754871711Richard Heylen
(14 Oct 2003)
49818548123107522887110525377542084788096962615971011
10610105 + 429312842269977506686435684442645814821681335988332440829Richard Heylen
(1 Aug 2003)
77867853716788858218366878517755270539950026013843817
10710107 - 569298108931508935388275225546217432865020464146509030899Richard Heylen
(22 Oct 2003)
335447849528797640933380616577564706751493530526472669
10810107 + 6969200017903493958497496269825834649922840822467385473869Richard Heylen
(30 Oct 2003)
499955245271433818753431743345545907675226421199679901
10910109 - 106172939293471451715027656817574156459818963532929889785991Richard Heylen
(30 Nov 2003)
3402178141490923281380298543164480324704096301595136513
11010109 + 190412762291535441394586840378227581982108373140647210683969David Cleaver
(16 Nov 2003)
3620182689515453682000526808729199471973438435837408289
11110111 - 1195714118203979842718112107179085913584270402578821772762747Richard Heylen
(5 Apr 2004)
70830539169695461569685870350892702678351054824192755569
11210111 + 5733916276909230503536141393381740491200391483078776043167501Sander Hoogendoorn
(15 Mar 2006)
61436725230731313013378040084827321449103017270012524839
11310113 - 91227135483731348216310599104139961914909156900674425643544869Sander Hoogendoorn
(20 Apr 2006)
738095998721668909879514364987164877946232346531808934017
11410113 + 29359129956782241538606843279942501870615823682988388592729389David Cleaver
(27 Mar 2007)
769486580655246475123803852893690968086716734490830614731
11510115 - 17691150412009677946969526346072331760233354867799371253396127David Cleaver
(1 Apr 2007)
8692537904571648356378342160363552634406563631760612220553
11610115 + 127411659340911132774307356923305106837591426034334709864681011Sander Hoogendoorn
(28 Apr 2006)
6026489151752033839345332361847936212205931651365703227431
11710117 - 273315726258283899741012641225377783011441520301423723971096511David Cleaver
(18 Dec 2007)
63587916588129671349443296973998097337310623412374598098797
11810117 + 734115050176328438144813324771965623501918056918556290873576027David Cleaver
(2 Oct 2008)
66444404249965125651345348476373415853054294809081436607383
11910119 - 19107106024525144985262742887581992250202679994837703390958334023David Cleaver
(16 Oct 2008)
943178003987785743709001406775672977264468496178170211877691
12010119 + 10443173351990368948770006118014625420408218097207571089955436713David Cleaver
(12 Apr 2010)
576860985484896071518016590923269597067312247472675097328211
12110121 - 96331367900569011726407096267890378427482659403198774965236311639David Cleaver
(13 Jan 2009)
7310472871010461886172597258203091860085059486059830493386553
12210121 + 74411605054949157064364983042540776217854522897157081818323775159David Cleaver
(27 Mar 2010)
6230316292443293371561630045207465493869335026340758295841399
12310123 - 2645328073200217603752002881036939514120445690511680371244497994669David Cleaver
(8 Apr 2010)
35621161543703660337594063098262226648277163072685110342932663
12410123 + 705123480802116351202045973315154645578302951428352231401178333959David Cleaver
(23 Apr 2010)
42587982942185577475731058591303564144452786494837956270583389
12510125 - 16557166570335815515196975472443830050489133849299043347272954825491David Cleaver
(27 Apr 2010)
600346991620133950473679222482756544469453098090964157383508673
12610125 + 4521132210627754162345171593168355643614439798921560464651676992659David Cleaver
(23 Oct 2012)
756368846428472780503267458846713228825124648050564337149416019
12710127 - 29073004614624254801141752548298266004216368150568442719640543520887David Cleaver
(24 Oct 2012)
3328213847884129619267433103503463111142404775417225305832697539
12810127 + 130091646512457786388220725173031177402273620251133069789682876014669David Cleaver
(27 Oct 2012)
6073443266529696079887253145345114140485378623176342153392307861
12910129 - 5594910064387855248848625711880039470176568628464546512725287405454369David Cleaver
(11 Nov 2012)
99360240720301050107567769406969213211673564194363561137033148179
13010129 + 12311664430848568631369629637442720892078329088847716754155425408347David Cleaver
(29 Apr 2010)
85730715281552914903964466215480820809680963827247781907539975009
13110131 - 92459197427054610275909503055573039977052804567984989369391427647811271David Cleaver
(28 Dec 2013)
506516192511718125201252481400143562457302882452455584996603306371
13510135 - 43120806007823760047177983366531024716689800868420623499605966016419309David Cleaver
(30 Apr 2010)
48063040659728094805747774345208478847915068326895332758975323553141
14010139 + 113891711572707277681401882653436576961984319628140860697745579201843022487David Cleaver
(14 May 2010)
5842579726516767990814692244466661242985404589259592495430696082753547
14510145 - 400672143224364496082970033572415797713808175407494841201819814719144014601703David Cleaver
(11 Mar 2011)
4665867076567697511894482542699566223786911354228186876669264728008220411
14810147 + 1911210754325987903848222358333498946062147584250342963305961590952868977237Andreas Höglund
(29 Jul 2006)
89200063699717094086741570686240243731188867201010229921060666406453209287
15010149 + 21553196175124573517092034922493422470908957491913295260473203518969764669282857Paul Leyland
(6 Jul 2006)
509748624946200879944668835201017220741864841301635474800405920115982269129
15510155 - 70301137483460682971758231017855816043699329165454253307288357133379781794162674443David Cleaver
(10 Sep 2012)
727360218481797830198550769882837374983036932333920470521165661950612603395593

Maximal and minimal 3-brilliant numbers

The next table contains the maximal and minimal numbers with given number of figures whose three prime factors have the same length when expressed in decimal notation.

Maximal and minimal 3-brilliant numbers
DigitsMinimalPrime factorsDiscovererMaximalPrime factorsDiscoverer
182
2
2
Trivial82
2
2
Richard Heylen
2122
2
3
982
7
7
Richard Heylen
31053
5
7
3437
7
7
Trivial
4133111
11
11
Trivial997113
13
59
Richard Heylen
51001317
19
31
9993737
37
73
Richard Heylen
610018117
71
83
91267397
97
97
Trivial
71030301101
101
101
Trivial107 - 293101
181
547
Richard Heylen
8107 + 127167
233
257
Sander Hoogendoorn108 - 333223
509
881
Richard Heylen
9108 + 727331
397
761
Sander Hoogendoorn991026973997
997
997
Trivial
10100931009
1009
1009
Trivial1010 - 5331117
1997
4483
Richard Heylen
111010 + 27971201
1367
6091
Richard Heylen1011 - 9933847
4481
5801
Richard Heylen
121011 + 7572879
4643
7481
Richard Heylen997339973
9973
9973
Trivial
1310007310007
10007
10007
Trivial1013 - 522910589
22679
41641
Richard Heylen
141013 + 49314779
22397
30211
Richard Heylen1014 - 588329333
53693
63493
Richard Heylen
151014 + 564336887
48679
55691
Richard Heylen99991399991
99991
99991
Trivial
161000033100003
100003
100003
Trivial1016 - 2903146383
182981
373339
Richard Heylen
171016 + 1251165833
205703
293149
Richard Heylen1017 - 7977251879
616999
643463
Richard Heylen
181017 + 649211801
589189
801341
Richard Heylen9999833999983
999983
999983
Trivial
19(106 + 3)31000003
1000003
1000003
Trivial1019 - 54171750769
1777931
3212597
Richard Heylen
201019 + 21491633561
2124037
2882057
Richard Heylen1020 - 112972994503
5625617
5936153
Richard Heylen
211020 + 60431418159
8250113
8547029
Richard Heylen(107 - 9)39999991
9999991
9999991
Trivial
22(107 + 19)310000019
10000019
10000019
Trivial1022 - 77310186663
19042489
51551861
Richard Heylen
231022 + 242710073153
31104971
31915729
Richard Heylen1023 - 48918775423
57543443
92558099
Richard Heylen
241023 + 150921110813
49732283
95248171
Richard Heylen(108 - 11)399999989
99999989
99999989
Trivial
25(108 + 7)3100000007
100000007
100000007
Trivial1025 - 3899125786989
158352281
502041889
Richard Heylen
261025 + 16189106371977
114768809
819122573
Richard Heylen1026 - 16359180617837
594451513
931371461
Richard Heylen
271026 + 5751236902847
554251279
761593127
Richard Heylen(109 - 63)3999999937
999999937
999999937
Trivial
28(109 + 7)31000000007
1000000007
1000000007
Trivial1028 - 130671292215579
1624184473
4764635399
Richard Heylen
291028 + 147071005841013
2737363219
3631936381
Richard Heylen1029 - 164092228847419
5521017059
8126444471
Richard Heylen
301029 + 290131691348917
6316788229
9359884141
Richard Heylen(1010 - 33)39999999967
9999999967
9999999967
Trivial
31(1010 + 19)310000000019
10000000019
10000000019
Trivial1031 - 6461113689527419
19896130049
36714949319
Richard Heylen
321031 + 589311198428391
13492646957
66182895839
Richard Heylen1032 - 2100341994700453
42965141327
55422902687
Richard Heylen
(26 Apr 2002)
331032 + 192726464641997
40362051277
93618299183
Richard Heylen(1011 - 23)399999999977
99999999977
99999999977
Trivial
34(1011 + 3)3100000000003
100000000003
100000000003
Trivial1034 - 14933114850721137
118686379657
733610250563
Richard Heylen
(26 Apr 2002)
351034 + 52737119495634017
224778984407
372299332423
Richard Heylen1035 - 25911273483275773
445466326247
820832127419
Richard Heylen
(26 Apr 2002)
361035 + 4399302832767711
345905509051
954640061059
Richard Heylen(1012 - 11)3999999999989
999999999989
999999999989
Trivial
37(1012 + 39)31000000000039
1000000000039
1000000000039
Trivial1037 - 288391262313101783
1946468711927
4069916348521
Richard Heylen
(26 Apr 2002)
381037 + 1238191236706881791
2374655713867
3405121160927
Jim Fougeron1038 - 1497593459465859009
5328480059843
5424848400043
Richard Heylen
391038 + 1580413586186059383
4367940094321
6383965165087
Jim Fougeron(1013 - 29)39999999999971
9999999999971
9999999999971
Trivial
40(1013 + 37)310000000000037
10000000000037
10000000000037
Trivial1040 - 21361717363198401447
20739646816249
27769554715361
Richard Heylen
411040 + 13095712814854302599
14964561505277
52146159843559
Richard Heylen1041 - 656734903487411119
42063334550063
68112603333689
Richard Heylen
421041 + 1735324407650660903
55069088040739
74398835289509
Richard Heylen(1014 - 27)399999999999973
99999999999973
99999999999973
Trivial
43(1014 + 31)3100000000000031
100000000000031
100000000000031
Trivial1043 - 146909172300505289967
211820845471373
273996333782401
Jim Fougeron
441043 + 59587101663821593379
154251196872133
637683274853741
Donovan Johnson
(17 Sep 2007)
1044 - 196587204797993723737
529220463976397
922651459305617
Jim Fougeron
451044 + 65863183984656831567
555439604761381
978546684274069
Jim Fougeron(1015 - 11)3999999999999989
999999999999989
999999999999989
Trivial
46(1015 + 37)31000000000000037
1000000000000037
1000000000000037
Trivial1046 - 655531257239303812309
1514462531713763
5251985486925041
Jim Fougeron
471046 + 505271379143216396571
2130751443448703
3402967810987379
Jim Fougeron1047 - 626732249340625367531
5634982736187193
7889549094998069
Jim Fougeron
481047 + 882912772015045385711
4080087583803271
8841682815196811
Jim Fougeron(1016 - 63)39999999999999937
9999999999999937
9999999999999937
Trivial
49(1016 + 61)310000000000000061
10000000000000061
10000000000000061
Trivial1049 - 19255710584419300262271
27959929042948027
33790677470564279
Jim Fougeron
501049 + 3174114778664291304861
25683013132372291
26346252045255691
Jim Fougeron1050 - 22945718946951128091531
63046191097099543
83714716546569371
Jim Fougeron
511050 + 12489121223204371030971
63048999783056603
74732729167701107
Jim Fougeron(1017 - 3)399999999999999997
99999999999999997
99999999999999997
Trivial
52(1017 + 3)3100000000000000003
100000000000000003
100000000000000003
Trivial1052 - 176477133998667582700839
174686342410900561
427209172135519637
Richard Heylen
(10 May 2002)
531052 + 136711127164238960370737
259430702647849189
303119329497323627
Richard Heylen
(10 May 2002)
1053 - 288257326171326139830487
479381522599043591
639547683418675679
Richard Heylen
(10 May 2002)
541053 + 17473124579059080641657
833903327864200817
962585347616078617
Richard Heylen
(10 May 2002)
(1018 - 11)3999999999999999989
999999999999999989
999999999999999989
Trivial
55(1018 + 3)31000000000000000003
1000000000000000003
1000000000000000003
Trivial1055 - 2195031575853845858412391
1655415652335968641
3833337050952127687
Richard Heylen
(14 May 2002)
561055 + 320891267227113342116519
1843431668851243501
4280736536975381531
Richard Heylen
(14 May 2002)
1056 - 406672395810946112780733
4466726666846584747
9344543235276726683
Richard Heylen
(14 May 2002)
571056 + 4926492514380642884427987
4342954421922749933
9157642889778573119
Richard Heylen
(14 May 2002)
(1019 - 39)39999999999999999961
9999999999999999961
9999999999999999961
Trivial
58(1019 + 51)310000000000000000051
10000000000000000051
10000000000000000051
Trivial1058 - 26723911632532594322798479
26618303856556487219
32295747692831078861
Richard Heylen
(22 May 2002)
591058 + 10808113828075410413032133
17980427028526107161
40219647399720069637
Richard Heylen
(22 May 2002)
1059 - 7661928256759918246059307
38549172961130847991
91804204822980213913
Richard Heylen
(22 May 2002)
601059 + 7610725996429356478022461
54578705970898911931
70479540545693881477
Richard Heylen
(22 May 2002)
(1020 - 11)399999999999999999989
99999999999999999989
99999999999999999989
Trivial
61(1020 + 39)3100000000000000000039
100000000000000000039
100000000000000000039
Trivial1061 - 8399140881594246452086939
238903948375968285643
297113520291914005513
Richard Heylen
(28 Jun 2002)
621061 + 188527198215253418203418427
212063210537599457801
237901727598652088501
Donovan Johnson
(27 Sep 2007)
1062 - 28967126765938565324144143
870031810053723383309
906697236767450667059
Richard Heylen
(27 Jun 2002)
631062 + 1058637295938758296861970419
448144368785742725801
754015388102253507623
Richard Heylen
(27 Jun 2002)
(1021 - 101)3999999999999999999899
999999999999999999899
999999999999999999899
Trivial
64(1021 + 117)31000000000000000000117
1000000000000000000117
1000000000000000000117
Trivial1064 - 576511061035345578788338541
1391023341645011752009
6775412341858197416921
Richard Heylen
(27 Jun 2002)
651064 + 858031714618450864782471611
2329234269479119539521
2503913582603801962513
Richard Heylen
(27 Jun 2002)
1065 - 17297513677679547530663653161
4254537295896375206353
6391073032088863400753
Richard Heylen
(30 Jan 2003)
661065 + 1492591835808612551245318829
5728537695516908168719
9508868068476315618809
Richard Heylen
(30 Jan 2003)
(1022 - 27)39999999999999999999973
9999999999999999999973
9999999999999999999973
Trivial
67(1022 + 9)310000000000000000000009
10000000000000000000009
10000000000000000000009
Trivial1067 - 90182110101615893692369271767
17531974986675607157429
56464866654909501084353
Richard Heylen
(18 May 2003)
681067 + 53403711589548025741092487547
18965422792905017360051
45495764971886030329621
Richard Heylen
(16 Mar 2003)
1068 - 2603912074017005366544874783
87236000943598050131963
94940708489892063116309
Richard Heylen
(5 Jan 2003)
691068 + 19250717663201672959682401741
68347532202803380194037
82833828425856284929771
Richard Heylen
(21 Mar 2003)
(1023 - 23)399999999999999999999977
99999999999999999999977
99999999999999999999977
Trivial
70(1023 + 117)3100000000000000000000117
100000000000000000000117
100000000000000000000117
Trivial1070 - 96509168408859236958014948709
240549351041680305101809
246848718161192945885111
Richard Heylen
(17 Mar 2003)
711070 + 438499125932443710597528749973
219951245631973461246409
361023891574669178861407
Richard Heylen
(14 Oct 2003)
1071 - 171350919411515989914399433
456095644213453949493941
624793759241398759715993
Richard Heylen
721071 + 283093223870743276389461972199
635533664645795858603047
702852345491672800196581
Richard Heylen
(14 Oct 2003)
(1024 - 257)3999999999999999999999743
999999999999999999999743
999999999999999999999743
Trivial
73(1024 + 7)31000000000000000000000007
1000000000000000000000007
1000000000000000000000007
Trivial1073 - 5011131084484584660005964247077
1546319684139286596653819
5963171949495925649715649
Donovan Johnson
(6 Oct 2007)
741073 + 1114411560742180771083281388349
1793577197271133261268279
3572306866130364500807171
Richard Heylen
(5 Apr 2004)
1074 - 235233146025203611921936230217
4492220097138647112441799
7075819992450556090832419
Richard Heylen
(5 Apr 2004)
751074 + 150614283452594120204339048033
4505070643133323591195913
5182083775943267656500109
Richard Heylen
(19 Feb 2004)
(1025 - 123)39999999999999999999999877
9999999999999999999999877
9999999999999999999999877
Trivial
76(1025 + 13)310000000000000000000000013
10000000000000000000000013
10000000000000000000000013
Trivial1076 - 29124915633890400259856213478307
23821644499003257798605369
26851045330260867259821197
Donovan Johnson
(9 Oct 2007)
771076 + 6531110609639225498094253225569
11880458198152686560316113
79335249741846477883350863
Donovan Johnson
(6 Oct 2007)
1077 - 91047336506267167202217741394931
51431759388815828667418817
53260003564945654181942701
Predrag Minovic
(8 Apr 2007)
781077 + 39535326022738012542679467808233
50789395724565737472464827
75661328886633682119158083
Donovan Johnson
(9 oct 2007)
(1026 - 141)399999999999999999999999859
99999999999999999999999859
99999999999999999999999859
Trivial
79(1026 + 67)3100000000000000000000000067
100000000000000000000000067
100000000000000000000000067
Trivial1079 - 392931143292513957372727386413437
264088185130417093421531701
264257624023096159956859837
Donovan Johnson
(16 Dec 2007)
801079 + 141007131192965933337080026159611
168138094305799472246027887
453339257290198969207621651
Sander Hoogendoorn
(30 Jan 2009)
1080 - 939867320710445259749682508487803
337386274565537430125750899
924186210928835649321701989
Predrag Minovic
(3 Sep 2012)
811080 + 512791130779613765521449101299013
788209302311915336992164037
970104219469530811374862511
Predrag Minovic
(12 Sep 2012)
(1027 - 99)3999999999999999999999999901
999999999999999999999999901
999999999999999999999999901
Trivial
82(1027 + 103)31000000000000000000000000103
1000000000000000000000000103
1000000000000000000000000103
Trivial1082 - 8246991401353720156326275741192121
1882572839967422997259954433
3790534383496722984007117357
Predrag Minovic
(2 Oct 2012)
831082 + 2712991211211090341386815274647791
1756474009918768587793772707
4700439132256912172773687727
Sander Hoogendoorn
(23 May 2013)
1083 - 8854231478505156238086532862154217
7852056384285210248669321399
8613779435343006682061087519
Predrag Minovic
(15 Jan 2014)

Maximal and minimal 4-brilliant numbers

The next table contains the maximal and minimal numbers with given number of figures whose four prime factors have the same length when expressed in decimal notation.

Maximal and minimal 4-brilliant numbers
DigitsMinimalPrime factorsDiscovererMaximalPrime factorsDiscoverer
216
  • 2
  • 2
  • 2
  • 2
Bouk de Water 90
  • 2
  • 3
  • 3
  • 5
Bouk de Water
3100
  • 2
  • 2
  • 5
  • 5
Bouk de Water 875
  • 5
  • 5
  • 5
  • 7
Bouk de Water
41029
  • 3
  • 7
  • 7
  • 7
Bouk de Water 2401
  • 7
  • 7
  • 7
  • 7
Trivial
514641
  • 11
  • 11
  • 11
  • 11
Trivial 99671
  • 11
  • 13
  • 17
  • 41
Bouk de Water
6100529
  • 11
  • 13
  • 19
  • 37
Bouk de Water 999973
  • 13
  • 13
  • 61
  • 97
Bouk de Water
7106 + 109
  • 11
  • 23
  • 59
  • 67
Bouk de Water 107 - 8709
  • 17
  • 73
  • 83
  • 97
Bouk de Water
8107 + 5647
  • 19
  • 61
  • 89
  • 97
Richard Heylen
(16 May 2002)
974
  • 97
  • 97
  • 97
  • 97
Trivial
91014
  • 101
  • 101
  • 101
  • 101
Trivial 109 - 2717
  • 107
  • 109
  • 179
  • 479
Bouk de Water
10109 + 2847
  • 107
  • 167
  • 191
  • 293
Bouk de Water 1010 - 1671
  • 163
  • 241
  • 277
  • 919
Bouk de Water
111010 + 861
  • 103
  • 157
  • 653
  • 947
Bouk de Water 1011 - 2289
  • 311
  • 397
  • 829
  • 977
Richard Heylen
(16 May 2002)
121011 + 24219
  • 389
  • 401
  • 643
  • 997
Bouk de Water 9974
  • 997
  • 997
  • 997
  • 997
Trivial
1310094
  • 1009
  • 1009
  • 1009
  • 1009
Trivial 1013 - 191
  • 1051
  • 1117
  • 2063
  • 4129
Bouk de Water
141013 + 54129
  • 1321
  • 1423
  • 2087
  • 2549
Bouk de Water 1014 - 11021
  • 2161
  • 2297
  • 3019
  • 6673
Bouk de Water
151014 + 9729
  • 1033
  • 3109
  • 5479
  • 5683
Bouk de Water 1015 - 8459
  • 2939
  • 4957
  • 7907
  • 8681
Bouk de Water
161015 + 11799
  • 3499
  • 5741
  • 6133
  • 8117
Bouk de Water 99734
  • 9973
  • 9973
  • 9973
  • 9973
Trivial
17100074
  • 10007
  • 10007
  • 10007
  • 10007
Trivial 1017 - 194271
  • 13337
  • 14537
  • 17401
  • 29641
Bouk de Water
181017 + 35241
  • 11717
  • 13469
  • 18047
  • 35111
Richard Heylen
(16 May 2002)
1018 - 31247
  • 15773
  • 19301
  • 40697
  • 80713
Richard Heylen
(16 May 2002)
191018 + 12861
  • 20327
  • 26849
  • 27431
  • 66797
Richard Heylen
(16 May 2002)
1019 - 165507
  • 31891
  • 38453
  • 83617
  • 97523
Richard Heylen
(16 May 2002)
201019 + 80101
  • 27103
  • 63317
  • 68213
  • 85427
Richard Heylen
(16 May 2002)
999914
  • 99991
  • 99991
  • 99991
  • 99991
Trivial
211000034
  • 100003
  • 100003
  • 100003
  • 100003
Trivial 1021 - 40167
  • 107453
  • 167747
  • 179497
  • 309079
Richard Heylen
(16 May 2002)
221021 + 200239
  • 100057
  • 123923
  • 164429
  • 490481
Richard Heylen
(16 May 2002)
1022 - 43197
  • 171761
  • 296827
  • 439493
  • 446293
Richard Heylen
(16 May 2002)
231022 + 146433
  • 140477
  • 272759
  • 309157
  • 844183
Richard Heylen
(16 May 2002)
1023 - 121227
  • 393629
  • 519031
  • 625909
  • 782003
Richard Heylen
(16 May 2002)
241023 + 22597
  • 304553
  • 460841
  • 765503
  • 930763
Richard Heylen
(16 May 2002)
9999834
  • 999983
  • 999983
  • 999983
  • 999983
Trivial
25(106 + 3)4
  • 1000003
  • 1000003
  • 1000003
  • 1000003
Trivial 1025 - 225371
  • 1090409
  • 1356721
  • 1794679
  • 3766459
Richard Heylen
(16 May 2002)
261025 + 719407
  • 1178231
  • 1184987
  • 1761671
  • 4065661
Richard Heylen
(16 May 2002)
1026 - 12227
  • 1494659
  • 2763377
  • 4873753
  • 4967687
Richard Heylen
(16 May 2002)
271026 + 44047
  • 1157759
  • 2399869
  • 4320469
  • 8330353
Richard Heylen
(16 May 2002)
1027 - 181239
  • 4668211
  • 5065199
  • 5361887
  • 7887427
Richard Heylen
(22 May 2002)
281027 + 358173
  • 2410043
  • 4761727
  • 9033361
  • 9646313
Richard Heylen
(22 May 2002)
(107 - 9)4
  • 9999991
  • 9999991
  • 9999991
  • 9999991
Trivial
29(107 + 19)4
  • 10000019
  • 10000019
  • 10000019
  • 10000019
Trivial 1029 - 121587
  • 10211363
  • 14281931
  • 22793899
  • 30082279
Sander Hoogendoorn
301029 + 784329
  • 13503449
  • 15690811
  • 21475043
  • 21977377
Jim Fougeron 1030 - 225131
  • 14671121
  • 16358479
  • 60823691
  • 68504801
Sander Hoogendoorn
311030 + 46197
  • 24847783
  • 26380859
  • 37443787
  • 40742123
Jim Fougeron 1031 - 64409
  • 43520599
  • 50243803
  • 63662927
  • 71834989
Sander Hoogendoorn
321031 + 120531
  • 40738811
  • 52961729
  • 56976001
  • 81346249
Jim Fougeron (108 - 11)4
  • 99999989
  • 99999989
  • 99999989
  • 99999989
Trivial
33(108 + 7)4
  • 100000007
  • 100000007
  • 100000007
  • 100000007
Trivial 1033 - 2981361
  • 138811943
  • 150388417
  • 172824791
  • 277174159
Richard Heylen
(9 Aug 2002)
341033 + 6379
  • 121022911
  • 132294061
  • 169058819
  • 369448771
Jim Fougeron 1034 - 291003
  • 160575161
  • 169865251
  • 543834983
  • 674139769
Richard Heylen
(9 Aug 2002)
351034 + 1167691
  • 117930893
  • 237943709
  • 381975661
  • 932958863
Jim Fougeron 1035 - 7133483
  • 249412637
  • 577688777
  • 800184277
  • 867356429
Richard Heylen
(9 Aug 2002)
361035 + 1553517
  • 229833959
  • 596374529
  • 837572501
  • 871052447
Jim Fougeron (109 - 63)4
  • 999999937
  • 999999937
  • 999999937
  • 999999937
Trivial
37(109 + 7)4
  • 1000000007
  • 1000000007
  • 1000000007
  • 1000000007
Trivial 1037 - 3392961
  • 1024919407
  • 1205624261
  • 2704285817
  • 2992579621
Jim Fougeron
381037 + 73269
  • 1015004657
  • 2113986011
  • 2124717653
  • 2193454499
Jim Fougeron 1038 - 1209213
  • 1277941309
  • 3496358689
  • 4329118921
  • 5169798247
Jim Fougeron
391038 + 1505619
  • 2535959857
  • 3100290539
  • 3280439369
  • 3877244737
Jim Fougeron 1039 - 716501
  • 2188752617
  • 6832015219
  • 7332337547
  • 9120361379
Jim Fougeron
401039 + 5560879
  • 2806287299
  • 5684681111
  • 7330103057
  • 8551684723
Jim Fougeron (1010 - 33)4
  • 9999999967
  • 9999999967
  • 9999999967
  • 9999999967
Trivial
41(1010 + 19)4
  • 10000000019
  • 10000000019
  • 10000000019
  • 10000000019
Trivial 1041 - 1752149
  • 11665636553
  • 12020765051
  • 12486083933
  • 57112766749
Jim Fougeron
421041 + 270387
  • 10057723799
  • 14695606843
  • 16439060371
  • 41156248421
Jim Fougeron 1042 - 1968921
  • 16387356763
  • 29844439783
  • 35069159969
  • 58304529779
Jim Fougeron
431042 + 190281
  • 12904941277
  • 28529199667
  • 51682064947
  • 52555058197
Jim Fougeron 1043 - 812357
  • 32266433293
  • 66076452031
  • 68130819217
  • 68842829513
Jim Fougeron
441043 + 186541
  • 36717661187
  • 40254606487
  • 71613321583
  • 94474714183
Jim Fougeron (1011 - 23)4
  • 99999999977
  • 99999999977
  • 99999999977
  • 99999999977
Trivial
45(1011 + 3)4
  • 100000000003
  • 100000000003
  • 100000000003
  • 100000000003
Trivial 1045 - 1021013
  • 111193045207
  • 162288480071
  • 170369552917
  • 325269018463
Phil Carmody
(26 Oct 2003)
461045 + 5945727
  • 109637540401
  • 130495801231
  • 157056159707
  • 445029913331
Richard Heylen
(11 Jul 2002)
1046 - 550623
  • 114257545709
  • 301659452561
  • 369064621643
  • 786132506111
Phil Carmody
(26 Oct 2003)
471046 + 3940503
  • 118764538069
  • 214272532333
  • 513125859067
  • 765813137317
Richard Heylen
(19 Jul 2002)
1047 - 12394547
  • 258602566111
  • 537704923703
  • 784384962709
  • 916840728049
Phil Carmody
(27 Oct 2003)
481047 + 6265441
  • 255256467691
  • 678739441577
  • 705264620759
  • 818404596557
Donovan Johnson
(30 Aug 2007)
(1012 - 11)4
  • 999999999989
  • 999999999989
  • 999999999989
  • 999999999989
Trivial
49(1012 + 39)4
  • 1000000000039
  • 1000000000039
  • 1000000000039
  • 1000000000039
Trivial 1049 - 1328811
  • 1123363694533
  • 1132900299593
  • 1619376154261
  • 4852216658021
Predrag Minovic
(25 Apr 2006)
501049 + 679663
  • 1159788556469
  • 1175245695677
  • 1976861376901
  • 3711216539651
Richard Heylen
(2 Aug 2002)
1050 - 1827771
  • 1131830822071
  • 2299071133453
  • 4626382993013
  • 8306622844691
Andreas Höglund
(22 Aug 2006)
511050 + 874531
  • 2031459510449
  • 2646219932437
  • 3821581909133
  • 4867687850339
Sander Hoogendoorn
(17 Nov 2003)
1051 - 2839931
  • 2610618693979
  • 4981318597973
  • 8607852782717
  • 8933412874271
Andreas Höglund
(16 Sep 2006)
521051 + 24031807
  • 3586045996369
  • 5155570211641
  • 6706659821857
  • 8064940359719
Andreas Höglund
(16 Sep 2006)
(1013 - 29)4
  • 9999999999971
  • 9999999999971
  • 9999999999971
  • 9999999999971
Trivial
53(1013 + 37)4
  • 10000000000037
  • 10000000000037
  • 10000000000037
  • 10000000000037
Trivial 1053 - 853581
  • 13801633011217
  • 15102457237171
  • 15359336773961
  • 31235570830697
Richard Heylen
541053 + 2626077
  • 12517623121403
  • 12777073989481
  • 16667362777847
  • 37512830115337
Donovan Johnson
(30 Aug 2007)
1054 - 5526581
  • 15039932275681
  • 28441324348441
  • 44484945266719
  • 52552241384381
Donovan Johnson
(26 Sep 2007)
551054 + 1293679
  • 15739418315123
  • 31371181661341
  • 38173288431427
  • 53054334421939
Donovan Johnson
(7 Sep 2007)
1055 - 1372949
  • 19993325988847
  • 73485361492133
  • 80952155039507
  • 84078644846843
Donovan Johnson
(26 Sep 2007)
561055 + 26082931
  • 31465199992697
  • 50998056344693
  • 66759528591367
  • 93347484189833
Donovan Johnson
(3 Oct 2007)
(1014 - 27)4
  • 99999999999973
  • 99999999999973
  • 99999999999973
  • 99999999999973
Trivial
57(1014 + 31)4
  • 100000000000031
  • 100000000000031
  • 100000000000031
  • 100000000000031
Trivial 1057 - 4541
  • 103086207782083
  • 152179523379973
  • 211276757911589
  • 301711242208009
Richard Heylen
(2 Jun 2003)
581057 + 1811253
  • 117856492897181
  • 131917327092977
  • 204532887090121
  • 314471646669889
Richard Heylen
(3 Sep 2002)
1058 - 4187751
  • 143428321659551
  • 236119351144679
  • 523981514063681
  • 563530643461601
Donovan Johnson
(26 Sep 2007)
591058 + 5803147
  • 122262486564571
  • 258372872730919
  • 555905946893789
  • 569453860946227
Donovan Johnson
(12 Sep 2007)
1059 - 14655741
  • 313736165207561
  • 536135329528819
  • 672035000546687
  • 884645142291223
Donovan Johnson
(26 Sep 2007)
601059 + 36711589
  • 266542160903959
  • 622245970511791
  • 630099845622983
  • 956891329848907
Donovan Johnson
(3 Oct 2007)
(1015 - 11)4
  • 999999999999989
  • 999999999999989
  • 999999999999989
  • 999999999999989
Trivial
61(1015 + 37)4
  • 1000000000000037
  • 1000000000000037
  • 1000000000000037
  • 1000000000000037
Trivial 1061 - 22336827
  • 1156112937602863
  • 1821568051966979
  • 1985228621423119
  • 2391904358614871
Donovan Johnson
(30 Sep 2007)
621061 + 470193
  • 1456513870035241
  • 1731482834726159
  • 1752389428378763
  • 2262749930928469
Donovan Johnson
(12 Sep 2007)
1062 - 4626803
  • 1622599674057919
  • 2172008813516923
  • 3073884742899533
  • 9230802373690957
Richard Heylen
(17 Sep 2003)
631062 + 5111959
  • 2279143986837019
  • 2779912264720531
  • 3576658593158251
  • 4412855031576581
Donovan Johnson
(12 Sep 2007)
1063 - 34575809
  • 3237564338664067
  • 3415229830176679
  • 9445468489873021
  • 9574986131281247
Donovan Johnson
(9 Oct 2007)
641063 + 18799597
  • 2659618573195897
  • 5339403019173071
  • 7570134814101641
  • 9302171161690691
Donovan Johnson
(17 Sep 2007)
(1016 - 63)4
  • 9999999999999937
  • 9999999999999937
  • 9999999999999937
  • 9999999999999937
Trivial
65(1016 + 61)4
  • 10000000000000061
  • 10000000000000061
  • 10000000000000061
  • 10000000000000061
Trivial 1065 - 11058639
  • 11991999226304927
  • 14191385939851301
  • 17850368215222619
  • 32918225659149497
Donovan Johnson
(3 Oct 2007)
661065 + 16528921
  • 11582582156494837
  • 16857241544984641
  • 19535307537073043
  • 26217293034303991
Donovan Johnson
(14 Sep 2007)
1066 - 1194869
  • 16689608733528161
  • 18877791141985067
  • 35471007677518727
  • 89480652326297719
Donovan Johnson
(30 Sep 2007)
671066 + 1037911
  • 21291651498132551
  • 27133413015756133
  • 33930229792704437
  • 51015180136202441
Donovan Johnson
(14 Sep 2007)
681067 + 3136153
  • 22337954099825591
  • 75478603724282501
  • 76572821197688237
  • 77456541233638159
Donovan Johnson
(17 Sep 2007)
(1017 - 3)4
  • 99999999999999997
  • 99999999999999997
  • 99999999999999997
  • 99999999999999997
Trivial
69(1017 + 3)4
  • 100000000000000003
  • 100000000000000003
  • 100000000000000003
  • 100000000000000003
Trivial
701069 + 2495233
  • 111053578017928633
  • 112251513276295871
  • 150800642421161699
  • 531951512758103069
Donovan Johnson
(14 Sep 2007)
711070 + 15754711
  • 145813808529168361
  • 205631780744229349
  • 389667325576323679
  • 855888358348768781
Donovan Johnson
(17 Sep 2007)
751074 + 5773243
  • 1375956653530253917
  • 2772933156399582559
  • 2963465354433379559
  • 8844147812502586159
Donovan Johnson
(26 Sep 2007)

Density of brilliant numbers (by Richard Heylen)

Instead of deriving density results using other people's work I decided to think about the problem myself for a change and I'm quite pleased with the results.

With brilliant numbers we're trying to represent a number in the region of b^n (where n is odd) as the product of two primes where the larger of the two primes ranges from b^(n/2) to b^((n+1)/2). First we work out the average number of factors that a number b^n+-x has in this range. If we say that the factor in the range b^((n-1)/2) to b^ (n/2) is p and the larger factor is q. It's easy to see that they have the same number of digits base b. If we plot p against q then the number of ways is the number of lattice points on the curve p= (b^n+x)/q in the relevant region. We can estimate this by looking at the area between the curves p=b^n/q and p=(b^n+1)/q between the specified limits. I make this to be roughly ln b /2 which, for base 10 is about 1.15.

I initially found it surprising that a number would be representable as the product of two numbers in the relevant ranges in on average more than one way. After all, some of these numbers are prime or are the product of a tiny prime and a large prime so I looked a bit closer.

So that's 19 numbers representable in a total of 22 ways which works out to an average of 1.158 which is reasonable agreement. We then have to make sure that both p and q are prime. I can make a handwaving argument for the probabilities being independent based on the lack of usable modularity constraints. Anyway, when p and q are similar magnitude roughly one in n^2/4 ln^2 b will both be prime this goes down(up?) to one in (n^2-1) ln^2 b when they are at the other extremes of their ranges. The difference of (ln b)^2 can be usefully neglected when n is large enough to be interesting.

This means that we have ln b /2 ways of which only one in n^2/4 ln^2 b are acceptable. This cancels nicely to yield a density of 2- brilliants of one in n^2/2 ln b.

This gives excellent agreement with computed densities for instance the density of the first 1000 base 3 2-brilliants greater than 3^57 is one in 1765.118 and 57^2/2 ln 3 =1784.7

We can therefore expect that the density of 2-brilliants near 10^95 will be about one in 10390. The factor of ln b in the formula completely explains the density's base dependence left unexplained by the previous analysis.

From a handwavey point of view you can try and explain the base dependence by observing that 2^95+x is smaller than 10^95+x but then one has to mumble incoherently about being more likely to split into similar sized factors and I can't really justify it.

I have done a theoretical analysis of the density of 3-brilliants.
For numbers b^n±x the proportion of 3-brilliants is 1 in (n^3 * ln b)/9 for n=2 mod 3 and 1 in 4*(n^3* ln b)/9 for n=1 mod 3.

If you have a number that can be in these tables or you have any comment please fill the form.

Last updated on May 28th, 2013.