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.
Maximal and minimal 2-brilliant numbers
Maximal and minimal base-2 2-brilliant numbers
Density of brilliant numbers (by Richard Heylen)
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.
| Digits | Minimal | Prime factors | Discoverer |
|---|---|---|---|
| 1 | 8 | 2 x 2 x 2 | Trivial |
| 2 | 12 | 2 x 2 x 3 | |
| 3 | 105 | 3 x 5 x 7 | |
| 4 | 1331 | 11 x 11 x 11 | Trivial |
| 5 | 10013 | 17 x 19 x 31 | |
| 6 | 100181 | 17 x 71 x 83 | |
| 7 | 1030301 | 101 x 101 x 101 | Trivial |
| 8 | 107 + 127 | 167 x 233 x 257 | Sander Hoogendoorn |
| 9 | 108 + 727 | 331 x 397 x 761 | Sander Hoogendoorn |
| 10 | 10093 | 1009 x 1009 x 1009 | Trivial |
| 11 | 1010 + 2797 | 1201 x 1367 x 6091 | Richard Heylen |
| 12 | 1011 + 757 | 2879 x 4643 x 7481 | Richard Heylen |
| 13 | 100073 | 10007 x 10007 x 10007 | Trivial |
| 14 | 1013 + 493 | 14779 x 22397 x 30211 | Richard Heylen |
| 15 | 1014 + 5643 | 36887 x 48679 x 55691 | Richard Heylen |
| 16 | 1000033 | 100003 x 100003 x 100003 | Trivial |
| 17 | 1016 + 1251 | 165833 x 205703 x 293149 | Richard Heylen |
| 18 | 1017 + 649 | 211801 x 589189 x 801341 | Richard Heylen |
| 19 | (106 + 3)3 | 1000003 x 1000003 x 1000003 | Trivial |
| 20 | 1019 + 2149 | 1633561 x 2124037 x 2882057 | Richard Heylen |
| 21 | 1020 + 6043 | 1418159 x 8250113 x 8547029 | Richard Heylen |
| 22 | (107 + 19)3 | 10000019 x 10000019 x 10000019 | Trivial |
| 23 | 1022 + 2427 | 10073153 x 31104971 x 31915729 | Richard Heylen |
| 24 | 1023 + 1509 | 21110813 x 49732283 x 95248171 | Richard Heylen |
| 25 | (108 + 7)3 | 100000007 x 100000007 x 100000007 | Trivial |
| 26 | 1025 + 16189 | 106371977 x 114768809 x 819122573 | Richard Heylen |
| 27 | 1026 + 5751 | 236902847 x 554251279 x 761593127 | Richard Heylen |
| 28 | (109 + 7)3 | 1000000007 x 1000000007 x 1000000007 | Trivial |
| 29 | 1028 + 14707 | 1005841013 x 2737363219 x 3631936381 | Richard Heylen |
| 30 | 1029 + 29013 | 1691348917 x 6316788229 x 9359884141 | Richard Heylen |
| 31 | (1010 + 19)3 | 1 0000000019 x 1 0000000019 x 1 0000000019 | Trivial |
| 32 | 1031 + 5893 | 1 1198428391 x 1 3492646957 x 6 6182895839 | Richard Heylen |
| 33 | 1032 + 1927 | 2 6464641997 x 4 0362051277 x 9 3618299183 | Richard Heylen |
| 34 | (1011 + 3)3 | 10 0000000003 x 10 0000000003 x 10 0000000003 | Trivial |
| 35 | 1034 + 52737 | 11 9495634017 x 22 4778984407 x 37 2299332423 | Richard Heylen |
| 36 | 1035 + 4399 | 30 2832767711 x 34 5905509051 x 95 4640061059 | Richard Heylen |
| 37 | (1012 + 39)3 | 100 0000000039 x 100 0000000039 x 100 0000000039 | Trivial |
| 38 | 1037 + 123819 | 123 6706881791 x 237 4655713867 x 340 5121160927 | Jim Fougeron |
| 39 | 1038 + 158041 | 358 6186059383 x 436 7940094321 x 638 3965165087 | Jim Fougeron |
| 40 | (1013 + 37)3 | 1000 0000000037 x 1000 0000000037 x 1000 0000000037 | Trivial |
| 41 | 1040 + 130957 | 1281 4854302599 x 1496 4561505277 x 5214 6159843559 | Richard Heylen |
| 42 | 1041 + 17353 | 2440 7650660903 x 5506 9088040739 x 7439 8835289509 | Richard Heylen |
| 43 | (1014 + 31)3 | 10000 0000000031 x 10000 0000000031 x 10000 0000000031 | Trivial |
| 44 | 1043 + 59587 | 10166 3821593379 x 15425 1196872133 x 63768 3274853741 | Donovan Johnson (17 Sep 2007) |
| 45 | 1044 + 65863 | 18398 4656831567 x 55543 9604761381 x 97854 6684274069 | Jim Fougeron |
| 46 | (1015 + 37)3 | 100000 0000000037 x 100000 0000000037 x 100000 0000000037 | Trivial |
| 47 | 1046 + 50527 | 137914 3216396571 x 213075 1443448703 x 340296 7810987379 | Jim Fougeron |
| 48 | 1047 + 88291 | 277201 5045385711 x 408008 7583803271 x 884168 2815196811 | Jim Fougeron |
| 49 | (1016 + 61)3 | 1000000 0000000061 x 1000000 0000000061 x 1000000 0000000061 | Trivial |
| 50 | 1049 + 31741 | 1477866 4291304861 x 2568301 3132372291 x 2634625 2045255691 | Jim Fougeron |
| 51 | 1050 + 124891 | 2122320 4371030971 x 6304899 9783056603 x 7473272 9167701107 | Jim Fougeron |
| 52 | (1017 + 3)3 | 10000000 0000000003 x 10000000 0000000003 x 10000000 0000000003 | Trivial |
| 53 | 1052 + 136711 | 12716423 8960370737 x 25943070 2647849189 x 30311932 9497323627 | Richard Heylen (10 May 2002) |
| 54 | 1053 + 17473 | 12457905 9080641657 x 83390332 7864200817 x 96258534 7616078617 | Richard Heylen (10 May 2002) |
| 55 | (1018 + 3)3 | 100000000 0000000003 x 100000000 0000000003 x 100000000 0000000003 | Trivial |
| 56 | 1055 + 32089 | 126722711 3342116519 x 184343166 8851243501 x 428073653 6975381531 | Richard Heylen (14 May 2002) |
| 57 | 1056 + 492649 | 251438064 2884427987 x 434295442 1922749933 x 915764288 9778573119 | Richard Heylen (14 May 2002) |
| 58 | (1019 + 51)3 | 1000000000 0000000051 x 1000000000 0000000051 x 1000000000 0000000051 | Trivial |
| 59 | 1058 + 108081 | 1382807541 0413032133 x 1798042702 8526107161 x 4021964739 9720069637 | Richard Heylen (22 May 2002) |
| 60 | 1059 + 76107 | 2599642935 6478022461 x 5457870597 0898911931 x 7047954054 5693881477 | Richard Heylen (22 May 2002) |
| 61 | (1020 + 39)3 | 1 0000000000 0000000039 x 1 0000000000 0000000039 x 1 0000000000 0000000039 | Trivial |
| 62 | 1061 + 188527 | 1 9821525341 8203418427 x 2 1206321053 7599457801 x 2 3790172759 8652088501 | Donovan Johnson (27 Sep 2007) |
| 63 | 1062 + 1058637 | 2 9593875829 6861970419 x 4 4814436878 5742725801 x 7 5401538810 2253507623 | Richard Heylen (27 Jun 2002) |
| 64 | (1021 + 117)3 | 10 0000000000 0000000117 x 10 0000000000 0000000117 x 10 0000000000 0000000117 | Trivial |
| 65 | 1064 + 85803 | 17 1461845086 4782471611 x 23 2923426947 9119539521 x 25 0391358260 3801962513 | Richard Heylen (27 Jun 2002) |
| 66 | 1065 + 149259 | 18 3580861255 1245318829 x 57 2853769551 6908168719 x 95 0886806847 6315618809 | Richard Heylen (30 Jan 2003) |
| 67 | (1022 + 9)3 | 100 0000000000 0000000009 x 100 0000000000 0000000009 x 100 0000000000 0000000009 | Trivial |
| 68 | 1067 + 534037 | 115 8954802574 1092487547 x 189 6542279290 5017360051 x 454 9576497188 6030329621 | Richard Heylen (16 Mar 2003) |
| 69 | 1068 + 192507 | 176 6320167295 9682401741 x 683 4753220280 3380194037 x 828 3382842585 6284929771 | Richard Heylen (21 Mar 2003) |
| 70 | (1023 + 117)3 | 1000 0000000000 0000000117 x 1000 0000000000 0000000117 x 1000 0000000000 0000000117 | Trivial |
| 71 | 1070 + 438499 | 1259 3244371059 7528749973 x 2199 5124563197 3461246409 x 3610 2389157466 9178861407 | Richard Heylen (14 Oct 2003) |
| 72 | 1071 + 283093 | 2238 7074327638 9461972199 x 6355 3366464579 5858603047 x 7028 5234549167 2800196581 | Richard Heylen (14 Oct 2003) |
| 73 | (1024 + 7)3 | 10000 0000000000 0000000007 x 10000 0000000000 0000000007 x 10000 0000000000 0000000007 | Trivial |
| 74 | 1073 + 111441 | 15607 4218077108 3281388349 x 17935 7719727113 3261268279 x 35723 0686613036 4500807171 | Richard Heylen (5 Apr 2004) |
| 75 | 1074 + 15061 | 42834 5259412020 4339048033 x 45050 7064313332 3591195913 x 51820 8377594326 7656500109 | Richard Heylen (19 Feb 2004) |
| 76 | (1025 + 13)3 | 100000 0000000000 0000000013 x 100000 0000000000 0000000013 x 100000 0000000000 0000000013 | Trivial |
| 77 | 1076 + 65311 | 106096 3922549809 4253225569 x 118804 5819815268 6560316113 x 793352 4974184647 7883350863 | Donovan Johnson (6 Oct 2007) |
| 78 | 1077 + 395353 | 260227 3801254267 9467808233 x 507893 9572456573 7472464827 x 756613 2888663368 2119158083 | Donovan Johnson (9 oct 2007) |
| 79 | (1026 + 67)3 | 1000000 0000000000 0000000067 x 1000000 0000000000 0000000067 x 1000000 0000000000 0000000067 | Trivial |
| 80 | 1079 + 141007 | 1311929 6593333708 0026159611 x 1681380 9430579947 2246027887 x 4533392 5729019896 9207621651 | Sander Hoogendoorn (30 Jan 2009) |
| 81 | 1080 + 512791 | 1307796 1376552144 9101299013 x 7882093 0231191533 6992164037 x 9701042 1946953081 1374862511 | Predrag Minovic (12 Sep 2012) |
| 82 | (1027 + 103)3 | 10000000 0000000000 0000000103 x 10000000 0000000000 0000000103 x 10000000 0000000000 0000000103 | Trivial |
| 83 | 1082 + 271299 | 12112110 9034138681 5274647791 x 17564740 0991876858 7793772707 x 47004391 3225691217 2773687727 | Sander Hoogendoorn (23 May 2013) |
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.
| Digits | Maximal | Prime factors | Discoverer |
|---|---|---|---|
| 1 | 8 | 2 x 2 x 2 | Richard Heylen |
| 2 | 98 | 2 x 7 x 7 | Richard Heylen |
| 3 | 343 | 7 x 7 x 7 | Trivial |
| 4 | 9971 | 13 x 13 x 59 | Richard Heylen |
| 5 | 99937 | 37 x 37 x 73 | Richard Heylen |
| 6 | 912673 | 97 x 97 x 97 | Trivial |
| 7 | 107 - 293 x | 101 x 181 x 547 | Richard Heylen |
| 8 | 108 - 333 x | 223 x 509 x 881 | Richard Heylen |
| 9 | 991026973 | 997 x 997 x 997 | Trivial |
| 10 | 1010 - 533 | 1117 x 1997 x 4483 | Richard Heylen |
| 11 | 1011 - 993 | 3847 x 4481 x 5801 | Richard Heylen |
| 12 | 99733 | 9973 x 9973 x 9973 | Trivial |
| 13 | 1013 - 5229 | 10589 x 22679 x 41641 | Richard Heylen |
| 14 | 1014 - 5883 | 29333 x 53693 x 63493 | Richard Heylen |
| 15 | 999913 | 99991 x 99991 x 99991 | Trivial |
| 16 | 1016 - 2903 | 146383 x 182981 x 373339 | Richard Heylen |
| 17 | 1017 - 7977 | 251879 x 616999 x 643463 | Richard Heylen |
| 18 | 9999833 | 999983 x 999983 x 999983 | Trivial |
| 19 | 1019 - 5417 | 1750769 x 1777931 x 3212597 | Richard Heylen |
| 20 | 1020 - 11297 | 2994503 x 5625617 x 5936153 | Richard Heylen |
| 21 | (107 - 9)3 | 9999991 x 9999991 x 9999991 | Trivial |
| 22 | 1022 - 773 | 10186663 x 19042489 x 51551861 | Richard Heylen |
| 23 | 1023 - 489 | 18775423 x 57543443 x 92558099 | Richard Heylen |
| 24 | (108 - 11)3 | 99999989 x 99999989 x 99999989 | Trivial |
| 25 | 1025 - 3899 | 125786989 x 158352281 x 502041889 | Richard Heylen |
| 26 | 1026 - 16359 | 180617837 x 594451513 x 931371461 | Richard Heylen |
| 27 | (109 - 63)3 | 999999937 x 999999937 x 999999937 | Trivial |
| 28 | 1028 - 13067 | 1292215579 x 1624184473 x 4764635399 | Richard Heylen |
| 29 | 1029 - 16409 | 2228847419 x 5521017059 x 8126444471 | Richard Heylen |
| 30 | (1010 - 33)3 | 9999999967 x 9999999967 x 9999999967 | Trivial |
| 31 | 1031 - 64611 | 1 3689527419 x 1 9896130049 x 3 6714949319 | Richard Heylen |
| 32 | 1032 - 21003 | 4 1994700453 x 4 2965141327 x 5 5422902687 | Richard Heylen (26 Apr 2002) |
| 33 | (1011 - 23)3 | 9 9999999977 x 9 9999999977 x 9 9999999977 | Trivial |
| 34 | 1034 - 14933 | 11 4850721137 x 11 8686379657 x 73 3610250563 | Richard Heylen (26 Apr 2002) |
| 35 | 1035 - 25911 | 27 3483275773 x 44 5466326247 x 82 0832127419 | Richard Heylen (26 Apr 2002) |
| 36 | (1012 - 11)3 | 99 9999999989 x 99 9999999989 x 99 9999999989 | Trivial |
| 37 | 1037 - 28839 | 126 2313101783 x 194 6468711927 x 406 9916348521 | Richard Heylen (26 Apr 2002) |
| 38 | 1038 - 149759 | 345 9465859009 x 532 8480059843 x 542 4848400043 | Richard Heylen |
| 39 | (1013 - 29)3 | 999 9999999971 x 999 9999999971 x 999 9999999971 | Trivial |
| 40 | 1040 - 213617 | 1736 3198401447 x 2073 9646816249 x 2776 9554715361 | Richard Heylen |
| 41 | 1041 - 6567 | 3490 3487411119 x 4206 3334550063 x 6811 2603333689 | Richard Heylen |
| 42 | (1014 - 27)3 | 9999 9999999973 x 9999 9999999973 x 9999 9999999973 | Trivial |
| 43 | 1043 - 146909 | 17230 0505289967 x 21182 0845471373 x 27399 6333782401 | Jim Fougeron |
| 44 | 1044 - 196587 | 20479 7993723737 x 52922 0463976397 x 92265 1459305617 | Jim Fougeron |
| 45 | (1015 - 11)3 | 99999 9999999989 x 99999 9999999989 x 99999 9999999989 | Trivial |
| 46 | 1046 - 65553 | 125723 9303812309 x 151446 2531713763 x 525198 5486925041 | Jim Fougeron |
| 47 | 1047 - 62673 | 224934 0625367531 x 563498 2736187193 x 788954 9094998069 | Jim Fougeron |
| 48 | (1016 - 63)3 | 999999 9999999937 x 999999 9999999937 x 999999 9999999937 | Trivial |
| 49 | 1049 - 192557 | 1058441 9300262271 x 2795992 9042948027 x 3379067 7470564279 | Jim Fougeron |
| 50 | 1050 - 229457 | 1894695 1128091531 x 6304619 1097099543 x 8371471 6546569371 | Jim Fougeron |
| 51 | (1017 - 3)3 | 9999999 9999999997 x 9999999 9999999997 x 9999999 9999999997 | Trivial |
| 52 | 1052 - 176477 | 13399866 7582700839 x 17468634 2410900561 x 42720917 2135519637 | Richard Heylen (10 May 2002) |
| 53 | 1053 - 288257 | 32617132 6139830487 x 47938152 2599043591 x 63954768 3418675679 | Richard Heylen (10 May 2002) |
| 54 | (1018 - 11)3 | 99999999 9999999989 x 99999999 9999999989 x 99999999 9999999989 | Trivial |
| 55 | 1055 - 219503 | 157585384 5858412391 x 165541565 2335968641 x 383333705 0952127687 | Richard Heylen (14 May 2002) |
| 56 | 1056 - 40667 | 239581094 6112780733 x 446672666 6846584747 x 934454323 5276726683 | Richard Heylen (14 May 2002) |
| 57 | (1019 - 39)3 | 999999999 9999999961 x 999999999 9999999961 x 999999999 9999999961 | Trivial |
| 58 | 1058 - 267239 | 1163253259 4322798479 x 2661830385 6556487219 x 3229574769 2831078861 | Richard Heylen (22 May 2002) |
| 59 | 1059 - 76619 | 2825675991 8246059307 x 3854917296 1130847991 x 9180420482 2980213913 | Richard Heylen (22 May 2002) |
| 60 | (1020 - 11)3 | 9999999999 9999999989 x 9999999999 9999999989 x 9999999999 9999999989 | Trivial |
| 61 | 1061 - 8399 | 1 4088159424 6452086939 x 2 3890394837 5968285643 x 2 9711352029 1914005513 | Richard Heylen (28 Jun 2002) |
| 62 | 1062 - 28967 | 1 2676593856 5324144143 x 8 7003181005 3723383309 x 9 0669723676 7450667059 | Richard Heylen (27 Jun 2002) |
| 63 | (1021 - 101)3 | 9 9999999999 9999999899 x 9 9999999999 9999999899 x 9 9999999999 9999999899 | Trivial |
| 64 | 1064 - 57651 | 10 6103534557 8788338541 x 13 9102334164 5011752009 x 67 7541234185 8197416921 | Richard Heylen (27 Jun 2002) |
| 65 | 1065 - 1729751 | 36 7767954753 0663653161 x 42 5453729589 6375206353 x 63 9107303208 8863400753 | Richard Heylen (30 Jan 2003) |
| 66 | (1022 - 27)3 | 99 9999999999 9999999973 x 99 9999999999 9999999973 x 99 9999999999 9999999973 | Trivial |
| 67 | 1067 - 901821 | 101 0161589369 2369271767 x 175 3197498667 5607157429 x 564 6486665490 9501084353 | Richard Heylen (18 May 2003) |
| 68 | 1068 - 26039 | 120 7401700536 6544874783 x 872 3600094359 8050131963 x 949 4070848989 2063116309 | Richard Heylen (5 Jan 2003) |
| 69 | (1023 - 23)3 | 999 9999999999 9999999977 x 999 9999999999 9999999977 x 999 9999999999 9999999977 | Trivial |
| 70 | 1070 - 96509 | 1684 0885923695 8014948709 x 2405 4935104168 0305101809 x 2468 4871816119 2945885111 | Richard Heylen (17 Mar 2003) |
| 71 | 1071 - 171 | 3509 1941151598 9914399433 x 4560 9564421345 3949493941 x 6247 9375924139 8759715993 | Richard Heylen |
| 72 | (1024 - 257)3 | 9999 9999999999 9999999743 x 9999 9999999999 9999999743 x 9999 9999999999 9999999743 | Trivial |
| 73 | 1073 - 501113 | 10844 8458466000 5964247077 x 15463 1968413928 6596653819 x 59631 7194949592 5649715649 | Donovan Johnson (6 Oct 2007) |
| 74 | 1074 - 23523 | 31460 2520361192 1936230217 x 44922 2009713864 7112441799 x 70758 1999245055 6090832419 | Richard Heylen (5 Apr 2004) |
| 75 | (1025 - 123)3 | 99999 9999999999 9999999877 x 99999 9999999999 9999999877 x 99999 9999999999 9999999877 | Trivial |
| 76 | 1076 - 291249 | 156338 9040025985 6213478307 x 238216 4449900325 7798605369 x 268510 4533026086 7259821197 | Donovan Johnson (9 Oct 2007) |
| 77 | 1077 - 910473 | 365062 6716720221 7741394931 x 514317 5938881582 8667418817 x 532600 0356494565 4181942701 | Predrag Minovic (8 Apr 2007) |
| 78 | (1026 - 141)3 | 999999 9999999999 9999999859 x 999999 9999999999 9999999859 x 999999 9999999999 9999999859 | Trivial |
| 79 | 1079 - 392931 | 1432925 1395737272 7386413437 x 2640881 8513041709 3421531701 x 2642576 2402309615 9956859837 | Donovan Johnson (16 Dec 2007) |
| 80 | 1080 - 939867 | 3207104 4525974968 2508487803 x 3373862 7456553743 0125750899 x 9241862 1092883564 9321701989 | Predrag Minovic (3 Sep 2012) |
| 81 | (1027 - 99)3 | 9999999 9999999999 9999999901 x 9999999 9999999999 9999999901 x 9999999 9999999999 9999999901 | Trivial |
| 82 | 1082 - 824699 | 14013537 2015632627 5741192121 x 18825728 3996742299 7259954433 x 37905343 8349672298 4007117357 | Predrag Minovic (2 Oct 2012) |
| 83 | 1083 - 885423 | 14785051 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.