3-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

Minimal 3-brilliant numbers

Maximal 3-brilliant numbers

Minimal 4-brilliant numbers

Maximal 4-brilliant numbers

Maximal and minimal base-2 2-brilliant numbers

Density of brilliant numbers (by Richard Heylen)

Minimal 3-brilliant numbers

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

Minimal 3-brilliant numbers
DigitsMinimalPrime factorsDiscoverer
182 x
2 x
2
Trivial
2122 x
2 x
3
31053 x
5 x
7
4133111 x
11 x
11
Trivial
51001317 x
19 x
31
610018117 x
71 x
83
71030301101 x
101 x
101
Trivial
8107 + 127167 x
233 x
257
Sander Hoogendoorn
9108 + 727331 x
397 x
761
Sander Hoogendoorn
10100931009 x
1009 x
1009
Trivial
111010 + 27971201 x
1367 x
6091
Richard Heylen
121011 + 7572879 x
4643 x
7481
Richard Heylen
1310007310007 x
10007 x
10007
Trivial
141013 + 49314779 x
22397 x
30211
Richard Heylen
151014 + 564336887 x
48679 x
55691
Richard Heylen
161000033100003 x
100003 x
100003
Trivial
171016 + 1251165833 x
205703 x
293149
Richard Heylen
181017 + 649211801 x
589189 x
801341
Richard Heylen
19(106 + 3)31000003 x
1000003 x
1000003
Trivial
201019 + 21491633561 x
2124037 x
2882057
Richard Heylen
211020 + 60431418159 x
8250113 x
8547029
Richard Heylen
22(107 + 19)310000019 x
10000019 x
10000019
Trivial
231022 + 242710073153 x
31104971 x
31915729
Richard Heylen
241023 + 150921110813 x
49732283 x
95248171
Richard Heylen
25(108 + 7)3100000007 x
100000007 x
100000007
Trivial
261025 + 16189106371977 x
114768809 x
819122573
Richard Heylen
271026 + 5751236902847 x
554251279 x
761593127
Richard Heylen
28(109 + 7)31000000007 x
1000000007 x
1000000007
Trivial
291028 + 147071005841013 x
2737363219 x
3631936381
Richard Heylen
301029 + 290131691348917 x
6316788229 x
9359884141
Richard Heylen
31(1010 + 19)31 0000000019 x
1 0000000019 x
1 0000000019
Trivial
321031 + 58931 1198428391 x
1 3492646957 x
6 6182895839
Richard Heylen
331032 + 19272 6464641997 x
4 0362051277 x
9 3618299183
Richard Heylen
34(1011 + 3)310 0000000003 x
10 0000000003 x
10 0000000003
Trivial
351034 + 5273711 9495634017 x
22 4778984407 x
37 2299332423
Richard Heylen
361035 + 439930 2832767711 x
34 5905509051 x
95 4640061059
Richard Heylen
37(1012 + 39)3100 0000000039 x
100 0000000039 x
100 0000000039
Trivial
381037 + 123819123 6706881791 x
237 4655713867 x
340 5121160927
Jim Fougeron
391038 + 158041358 6186059383 x
436 7940094321 x
638 3965165087
Jim Fougeron
40(1013 + 37)31000 0000000037 x
1000 0000000037 x
1000 0000000037
Trivial
411040 + 1309571281 4854302599 x
1496 4561505277 x
5214 6159843559
Richard Heylen
421041 + 173532440 7650660903 x
5506 9088040739 x
7439 8835289509
Richard Heylen
43(1014 + 31)310000 0000000031 x
10000 0000000031 x
10000 0000000031
Trivial
441043 + 5958710166 3821593379 x
15425 1196872133 x
63768 3274853741
Donovan Johnson
(17 Sep 2007)
451044 + 6586318398 4656831567 x
55543 9604761381 x
97854 6684274069
Jim Fougeron
46(1015 + 37)3100000 0000000037 x
100000 0000000037 x
100000 0000000037
Trivial
471046 + 50527137914 3216396571 x
213075 1443448703 x
340296 7810987379
Jim Fougeron
481047 + 88291277201 5045385711 x
408008 7583803271 x
884168 2815196811
Jim Fougeron
49(1016 + 61)31000000 0000000061 x
1000000 0000000061 x
1000000 0000000061
Trivial
501049 + 317411477866 4291304861 x
2568301 3132372291 x
2634625 2045255691
Jim Fougeron
511050 + 1248912122320 4371030971 x
6304899 9783056603 x
7473272 9167701107
Jim Fougeron
52(1017 + 3)310000000 0000000003 x
10000000 0000000003 x
10000000 0000000003
Trivial
531052 + 13671112716423 8960370737 x
25943070 2647849189 x
30311932 9497323627
Richard Heylen
(10 May 2002)
541053 + 1747312457905 9080641657 x
83390332 7864200817 x
96258534 7616078617
Richard Heylen
(10 May 2002)
55(1018 + 3)3100000000 0000000003 x
100000000 0000000003 x
100000000 0000000003
Trivial
561055 + 32089126722711 3342116519 x
184343166 8851243501 x
428073653 6975381531
Richard Heylen
(14 May 2002)
571056 + 492649251438064 2884427987 x
434295442 1922749933 x
915764288 9778573119
Richard Heylen
(14 May 2002)
58(1019 + 51)31000000000 0000000051 x
1000000000 0000000051 x
1000000000 0000000051
Trivial
591058 + 1080811382807541 0413032133 x
1798042702 8526107161 x
4021964739 9720069637
Richard Heylen
(22 May 2002)
601059 + 761072599642935 6478022461 x
5457870597 0898911931 x
7047954054 5693881477
Richard Heylen
(22 May 2002)
61(1020 + 39)31 0000000000 0000000039 x
1 0000000000 0000000039 x
1 0000000000 0000000039
Trivial
621061 + 1885271 9821525341 8203418427 x
2 1206321053 7599457801 x
2 3790172759 8652088501
Donovan Johnson
(27 Sep 2007)
631062 + 10586372 9593875829 6861970419 x
4 4814436878 5742725801 x
7 5401538810 2253507623
Richard Heylen
(27 Jun 2002)
64(1021 + 117)310 0000000000 0000000117 x
10 0000000000 0000000117 x
10 0000000000 0000000117
Trivial
651064 + 8580317 1461845086 4782471611 x
23 2923426947 9119539521 x
25 0391358260 3801962513
Richard Heylen
(27 Jun 2002)
661065 + 14925918 3580861255 1245318829 x
57 2853769551 6908168719 x
95 0886806847 6315618809
Richard Heylen
(30 Jan 2003)
67(1022 + 9)3100 0000000000 0000000009 x
100 0000000000 0000000009 x
100 0000000000 0000000009
Trivial
681067 + 534037115 8954802574 1092487547 x
189 6542279290 5017360051 x
454 9576497188 6030329621
Richard Heylen
(16 Mar 2003)
691068 + 192507176 6320167295 9682401741 x
683 4753220280 3380194037 x
828 3382842585 6284929771
Richard Heylen
(21 Mar 2003)
70(1023 + 117)31000 0000000000 0000000117 x
1000 0000000000 0000000117 x
1000 0000000000 0000000117
Trivial
711070 + 4384991259 3244371059 7528749973 x
2199 5124563197 3461246409 x
3610 2389157466 9178861407
Richard Heylen
(14 Oct 2003)
721071 + 2830932238 7074327638 9461972199 x
6355 3366464579 5858603047 x
7028 5234549167 2800196581
Richard Heylen
(14 Oct 2003)
73(1024 + 7)310000 0000000000 0000000007 x
10000 0000000000 0000000007 x
10000 0000000000 0000000007
Trivial
741073 + 11144115607 4218077108 3281388349 x
17935 7719727113 3261268279 x
35723 0686613036 4500807171
Richard Heylen
(5 Apr 2004)
751074 + 1506142834 5259412020 4339048033 x
45050 7064313332 3591195913 x
51820 8377594326 7656500109
Richard Heylen
(19 Feb 2004)
76(1025 + 13)3100000 0000000000 0000000013 x
100000 0000000000 0000000013 x
100000 0000000000 0000000013
Trivial
771076 + 65311106096 3922549809 4253225569 x
118804 5819815268 6560316113 x
793352 4974184647 7883350863
Donovan Johnson
(6 Oct 2007)
781077 + 395353260227 3801254267 9467808233 x
507893 9572456573 7472464827 x
756613 2888663368 2119158083
Donovan Johnson
(9 oct 2007)
79(1026 + 67)31000000 0000000000 0000000067 x
1000000 0000000000 0000000067 x
1000000 0000000000 0000000067
Trivial
801079 + 1410071311929 6593333708 0026159611 x
1681380 9430579947 2246027887 x
4533392 5729019896 9207621651
Sander Hoogendoorn
(30 Jan 2009)
811080 + 5127911307796 1376552144 9101299013 x
7882093 0231191533 6992164037 x
9701042 1946953081 1374862511
Predrag Minovic
(12 Sep 2012)
82(1027 + 103)310000000 0000000000 0000000103 x
10000000 0000000000 0000000103 x
10000000 0000000000 0000000103
Trivial
831082 + 27129912112110 9034138681 5274647791 x
17564740 0991876858 7793772707 x
47004391 3225691217 2773687727
Sander Hoogendoorn
(23 May 2013)

Maximal 3-brilliant numbers

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

Maximal 3-brilliant numbers
DigitsMaximalPrime factorsDiscoverer
182 x
2 x
2
Richard Heylen
2982 x
7 x
7
Richard Heylen
33437 x
7 x
7
Trivial
4997113 x
13 x
59
Richard Heylen
59993737 x
37 x
73
Richard Heylen
691267397 x
97 x
97
Trivial
7107 - 293 x101 x
181 x
547
Richard Heylen
8108 - 333 x223 x
509 x
881
Richard Heylen
9991026973997 x
997 x
997
Trivial
101010 - 5331117 x
1997 x
4483
Richard Heylen
111011 - 9933847 x
4481 x
5801
Richard Heylen
12997339973 x
9973 x
9973
Trivial
131013 - 522910589 x
22679 x
41641
Richard Heylen
141014 - 588329333 x
53693 x
63493
Richard Heylen
1599991399991 x
99991 x
99991
Trivial
161016 - 2903146383 x
182981 x
373339
Richard Heylen
171017 - 7977251879 x
616999 x
643463
Richard Heylen
189999833999983 x
999983 x
999983
Trivial
191019 - 54171750769 x
1777931 x
3212597
Richard Heylen
201020 - 112972994503 x
5625617 x
5936153
Richard Heylen
21(107 - 9)39999991 x
9999991 x
9999991
Trivial
221022 - 77310186663 x
19042489 x
51551861
Richard Heylen
231023 - 48918775423 x
57543443 x
92558099
Richard Heylen
24(108 - 11)399999989 x
99999989 x
99999989
Trivial
251025 - 3899125786989 x
158352281 x
502041889
Richard Heylen
261026 - 16359180617837 x
594451513 x
931371461
Richard Heylen
27(109 - 63)3999999937 x
999999937 x
999999937
Trivial
281028 - 130671292215579 x
1624184473 x
4764635399
Richard Heylen
291029 - 164092228847419 x
5521017059 x
8126444471
Richard Heylen
30(1010 - 33)39999999967 x
9999999967 x
9999999967
Trivial
311031 - 646111 3689527419 x
1 9896130049 x
3 6714949319
Richard Heylen
321032 - 210034 1994700453 x
4 2965141327 x
5 5422902687
Richard Heylen
(26 Apr 2002)
33(1011 - 23)39 9999999977 x
9 9999999977 x
9 9999999977
Trivial
341034 - 1493311 4850721137 x
11 8686379657 x
73 3610250563
Richard Heylen
(26 Apr 2002)
351035 - 2591127 3483275773 x
44 5466326247 x
82 0832127419
Richard Heylen
(26 Apr 2002)
36(1012 - 11)399 9999999989 x
99 9999999989 x
99 9999999989
Trivial
371037 - 28839126 2313101783 x
194 6468711927 x
406 9916348521
Richard Heylen
(26 Apr 2002)
381038 - 149759345 9465859009 x
532 8480059843 x
542 4848400043
Richard Heylen
39(1013 - 29)3999 9999999971 x
999 9999999971 x
999 9999999971
Trivial
401040 - 2136171736 3198401447 x
2073 9646816249 x
2776 9554715361
Richard Heylen
411041 - 65673490 3487411119 x
4206 3334550063 x
6811 2603333689
Richard Heylen
42(1014 - 27)39999 9999999973 x
9999 9999999973 x
9999 9999999973
Trivial
431043 - 14690917230 0505289967 x
21182 0845471373 x
27399 6333782401
Jim Fougeron
441044 - 19658720479 7993723737 x
52922 0463976397 x
92265 1459305617
Jim Fougeron
45(1015 - 11)399999 9999999989 x
99999 9999999989 x
99999 9999999989
Trivial
461046 - 65553125723 9303812309 x
151446 2531713763 x
525198 5486925041
Jim Fougeron
471047 - 62673224934 0625367531 x
563498 2736187193 x
788954 9094998069
Jim Fougeron
48(1016 - 63)3999999 9999999937 x
999999 9999999937 x
999999 9999999937
Trivial
491049 - 1925571058441 9300262271 x
2795992 9042948027 x
3379067 7470564279
Jim Fougeron
501050 - 2294571894695 1128091531 x
6304619 1097099543 x
8371471 6546569371
Jim Fougeron
51(1017 - 3)39999999 9999999997 x
9999999 9999999997 x
9999999 9999999997
Trivial
521052 - 17647713399866 7582700839 x
17468634 2410900561 x
42720917 2135519637
Richard Heylen
(10 May 2002)
531053 - 28825732617132 6139830487 x
47938152 2599043591 x
63954768 3418675679
Richard Heylen
(10 May 2002)
54(1018 - 11)399999999 9999999989 x
99999999 9999999989 x
99999999 9999999989
Trivial
551055 - 219503157585384 5858412391 x
165541565 2335968641 x
383333705 0952127687
Richard Heylen
(14 May 2002)
561056 - 40667239581094 6112780733 x
446672666 6846584747 x
934454323 5276726683
Richard Heylen
(14 May 2002)
57(1019 - 39)3999999999 9999999961 x
999999999 9999999961 x
999999999 9999999961
Trivial
581058 - 2672391163253259 4322798479 x
2661830385 6556487219 x
3229574769 2831078861
Richard Heylen
(22 May 2002)
591059 - 766192825675991 8246059307 x
3854917296 1130847991 x
9180420482 2980213913
Richard Heylen
(22 May 2002)
60(1020 - 11)39999999999 9999999989 x
9999999999 9999999989 x
9999999999 9999999989
Trivial
611061 - 83991 4088159424 6452086939 x
2 3890394837 5968285643 x
2 9711352029 1914005513
Richard Heylen
(28 Jun 2002)
621062 - 289671 2676593856 5324144143 x
8 7003181005 3723383309 x
9 0669723676 7450667059
Richard Heylen
(27 Jun 2002)
63(1021 - 101)39 9999999999 9999999899 x
9 9999999999 9999999899 x
9 9999999999 9999999899
Trivial
641064 - 5765110 6103534557 8788338541 x
13 9102334164 5011752009 x
67 7541234185 8197416921
Richard Heylen
(27 Jun 2002)
651065 - 172975136 7767954753 0663653161 x
42 5453729589 6375206353 x
63 9107303208 8863400753
Richard Heylen
(30 Jan 2003)
66(1022 - 27)399 9999999999 9999999973 x
99 9999999999 9999999973 x
99 9999999999 9999999973
Trivial
671067 - 901821101 0161589369 2369271767 x
175 3197498667 5607157429 x
564 6486665490 9501084353
Richard Heylen
(18 May 2003)
681068 - 26039120 7401700536 6544874783 x
872 3600094359 8050131963 x
949 4070848989 2063116309
Richard Heylen
(5 Jan 2003)
69(1023 - 23)3999 9999999999 9999999977 x
999 9999999999 9999999977 x
999 9999999999 9999999977
Trivial
701070 - 965091684 0885923695 8014948709 x
2405 4935104168 0305101809 x
2468 4871816119 2945885111
Richard Heylen
(17 Mar 2003)
711071 - 1713509 1941151598 9914399433 x
4560 9564421345 3949493941 x
6247 9375924139 8759715993
Richard Heylen
72(1024 - 257)39999 9999999999 9999999743 x
9999 9999999999 9999999743 x
9999 9999999999 9999999743
Trivial
731073 - 50111310844 8458466000 5964247077 x
15463 1968413928 6596653819 x
59631 7194949592 5649715649
Donovan Johnson
(6 Oct 2007)
741074 - 2352331460 2520361192 1936230217 x
44922 2009713864 7112441799 x
70758 1999245055 6090832419
Richard Heylen
(5 Apr 2004)
75(1025 - 123)399999 9999999999 9999999877 x
99999 9999999999 9999999877 x
99999 9999999999 9999999877
Trivial
761076 - 291249156338 9040025985 6213478307 x
238216 4449900325 7798605369 x
268510 4533026086 7259821197
Donovan Johnson
(9 Oct 2007)
771077 - 910473365062 6716720221 7741394931 x
514317 5938881582 8667418817 x
532600 0356494565 4181942701
Predrag Minovic
(8 Apr 2007)
78(1026 - 141)3999999 9999999999 9999999859 x
999999 9999999999 9999999859 x
999999 9999999999 9999999859
Trivial
791079 - 3929311432925 1395737272 7386413437 x
2640881 8513041709 3421531701 x
2642576 2402309615 9956859837
Donovan Johnson
(16 Dec 2007)
801080 - 9398673207104 4525974968 2508487803 x
3373862 7456553743 0125750899 x
9241862 1092883564 9321701989
Predrag Minovic
(3 Sep 2012)
81(1027 - 99)39999999 9999999999 9999999901 x
9999999 9999999999 9999999901 x
9999999 9999999999 9999999901
Trivial
821082 - 82469914013537 2015632627 5741192121 x
18825728 3996742299 7259954433 x
37905343 8349672298 4007117357
Predrag Minovic
(2 Oct 2012)
831083 - 88542314785051 5623808653 2862154217 x
78520563 8428521024 8669321399 x
86137794 3534300668 2061087519
Predrag Minovic
(15 Jan 2014)

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

Last updated on May 31st, 2015.