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 four prime factors have the same length when expressed in decimal notation.
| Digits | Minimal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 16 | 2x 2 × 2 × 2 | Bouk de Water |
| 3 | 100 | 2 × 2 × 5 × 5 | Bouk de Water |
| 4 | 1029 | 3 × 7 × 7 × 7 | Bouk de Water |
| 5 | 14641 | 11 × 11 × 11 × 11 | Trivial |
| 6 | 100529 | 11 × 13 × 19 × 37 | Bouk de Water |
| 7 | 106 + 109 | 11 × 23 × 59 × 67 | Bouk de Water |
| 8 | 107 + 5647 | 19 × 61 × 89 × 97 | Richard Heylen (16 May 2002) |
| 9 | 1014 | 101 × 101 × 101 × 101 | Trivial |
| 10 | 109 + 2847 | 107 × 167 × 191 × 293 | Bouk de Water |
| 11 | 1010 + 861 | 103 × 157 × 653 × 947 | Bouk de Water |
| 12 | 1011 + 24219 | 389 × 401 × 643 × 997 | Bouk de Water |
| 13 | 10094 | 1009 × 1009 × 1009 × 1009 | Trivial |
| 14 | 1013 + 51429 | 1321 × 1423 × 2087 × 2549 | Bouk de Water |
| 15 | 1014 + 9729 | 1033 × 3109 × 5479 × 5683 | Bouk de Water |
| 16 | 1015 + 11799 | 3499 × 5741 × 6133 × 8117 | Bouk de Water |
| 17 | 100074 | 10007 × 10007 × 10007 × 10007 | Trivial |
| 18 | 1017 + 35241 | 11717 × 13469 × 18047 × 35111 | Richard Heylen (16 May 2002) |
| 19 | 1018 + 12861 | 20327 × 26849 × 27431 × 66797 | Richard Heylen (16 May 2002) |
| 20 | 1019 + 80101 | 27103 × 63317 × 68213 × 85427 | Richard Heylen (16 May 2002) |
| 21 | 1000034 | 100003 × 100003 × 100003 × 100003 | Trivial |
| 22 | 1021 + 200239 | 100057 × 123923 × 164429 × 490481 | Richard Heylen (16 May 2002) |
| 23 | 1022 + 146433 | 140477 × 272759 × 309157 × 844183 | Richard Heylen (16 May 2002) |
| 24 | 1023 + 22597 | 304553 × 460841 × 765503 × 930763 | Richard Heylen (16 May 2002) |
| 25 | (106 + 3)4 | 1000003 × 1000003 × 1000003 × 1000003 | Trivial |
| 26 | 1025 + 719407 | 1178231 × 1184987 × 1761671 × 4065661 | Richard Heylen (16 May 2002) |
| 27 | 1026 + 44047 | 1157759 × 2399869 × 4320469 × 8330353 | Richard Heylen (16 May 2002) |
| 28 | 1027 + 358173 | 2410043 × 4761727 × 9033361 × 9646313 | Richard Heylen (22 May 2002) |
| 29 | (107 + 19)4 | 10000019 × 10000019 × 10000019 × 10000019 | Trivial |
| 30 | 1029 + 784329 | 13503449 × 15690811 × 21475043 × 21977377 | Jim Fougeron |
| 31 | 1030 + 46197 | 24847783 × 26380859 × 37443787 × 40742123 | Jim Fougeron |
| 32 | 1031 + 120531 | 40738811 × 52961729 × 56976001 × 81346249 | Jim Fougeron |
| 33 | (108 + 7)4 | 100000007 × 100000007 × 100000007 × 100000007 | Trivial |
| 34 | 1033 + 6379 | 121022911 × 132294061 × 169058819 × 369448771 | Jim Fougeron |
| 35 | 1034 + 1167691 | 117930893 × 237943709 × 381975661 × 932958863 | Jim Fougeron |
| 36 | 1035 + 1553517 | 229833959 × 596374529 × 837572501 × 871052447 | Jim Fougeron |
| 37 | (109 + 7)4 | 1000000007 × 1000000007 × 1000000007 × 1000000007 | Trivial |
| 38 | 1037 + 73269 | 1015004657 × 2113986011 × 2124717653 × 2193454499 | Jim Fougeron |
| 39 | 1038 + 1505619 | 2535959857 × 3100290539 × 3280439369 × 3877244737 | Jim Fougeron |
| 40 | 1039 + 5560879 | 2806287299 × 5684681111 × 7330103057 × 8551684723 | Jim Fougeron |
| 41 | (1010 + 19)4 | 1 0000000019 × 1 0000000019 × 1 0000000019 × 1 0000000019 | Trivial |
| 42 | 1041 + 270387 | 1 0057723799 × 1 4695606843 × 1 6439060371 × 4 1156248421 | Jim Fougeron |
| 43 | 1042 + 190281 | 1 2904941277 × 2 8529199667 × 5 1682064947 × 5 2555058197 | Jim Fougeron |
| 44 | 1043 + 186541 | 3 6717661187 × 4 0254606487 × 7 1613321583 × 9 4474714183 | Jim Fougeron |
| 45 | (1011 + 3)4 | 10 0000000003 × 10 0000000003 × 10 0000000003 × 10 0000000003 | Trivial |
| 46 | 1045 + 5945727 | 10 9637540401 × 13 0495801231 × 15 7056159707 × 44 5029913331 | Richard Heylen (11 Jul 2002) |
| 47 | 1046 + 3940503 | 11 8764538069 × 21 4272532333 × 51 3125859067 × 76 5813137317 | Richard Heylen (19 Jul 2002) |
| 48 | 1047 + 6265441 | 25 5256467691 × 67 8739441577 × 70 5264620759 × 81 8404596557 | Donovan Johnson (30 Aug 2007) |
| 49 | (1012 + 39)4 | 100 0000000039 × 100 0000000039 × 100 0000000039 × 100 0000000039 | Trivial |
| 50 | 1049 + 679663 | 115 9788556469 × 117 5245695677 × 197 6861376901 × 371 1216539651 | Richard Heylen (2 Aug 2002) |
| 51 | 1050 + 874531 | 203 1459510449 × 264 6219932437 × 382 1581909133 × 486 7687850339 | Sander Hoogendoorn (17 Nov 2003) |
| 52 | 1051 + 24031807 | 358 6045996369 × 515 5570211641 × 670 6659821857 × 806 4940359719 | Andreas Höglund (16 Sep 2006) |
| 53 | (1013 + 37)4 | 1000 0000000037 × 1000 0000000037 × 1000 0000000037 × 1000 0000000037 | Trivial |
| 54 | 1053 + 2626077 | 1251 7623121403 × 1277 7073989481 × 1666 7362777847 × 3751 2830115337 | Donovan Johnson (30 Aug 2007) |
| 55 | 1054 + 1293679 | 1573 9418315123 × 3137 1181661341 × 3817 3288431427 × 5305 4334421939 | Donovan Johnson (7 Sep 2007) |
| 56 | 1055 + 26082931 | 3146 5199992697 × 5099 8056344693 × 6675 9528591367 × 9334 7484189833 | Donovan Johnson (3 Oct 2007) |
| 57 | (1014 + 31)4 | 10000 0000000031 × 10000 0000000031 × 10000 0000000031 × 10000 0000000031 | Trivial |
| 58 | 1057 + 1811253 | 11785 6492897181 × 13191 7327092977 × 20453 2887090121 × 31447 1646669889 | Richard Heylen (3 Sep 2002) |
| 59 | 1058 + 5803147 | 12226 2486564571 × 25837 2872730919 × 55590 5946893789 × 56945 3860946227 | Donovan Johnson (12 Sep 2007) |
| 60 | 1059 + 36711589 | 26654 2160903959 × 62224 5970511791 × 63009 9845622983 × 95689 1329848907 | Donovan Johnson (3 Oct 2007) |
| 61 | (1015 + 37)4 | 100000 0000000037 × 100000 0000000037 × 100000 0000000037 × 100000 0000000037 | Trivial |
| 62 | 1061 + 470193 | 145651 3870035241 × 173148 2834726159 × 175238 9428378763 × 226274 9930928469 | Donovan Johnson (12 Sep 2007) |
| 63 | 1062 + 5111959 | 227914 3986837019 × 277991 2264720531 × 357665 8593158251 × 441285 5031576581 | Donovan Johnson (12 Sep 2007) |
| 64 | 1063 + 18799597 | 265961 8573195897 × 533940 3019173071 × 757013 4814101641 × 930217 1161690691 | Donovan Johnson (17 Sep 2007) |
| 65 | (1016 + 61)4 | 1000000 0000000061 × 1000000 0000000061 × 1000000 0000000061 × 1000000 0000000061 | Trivial |
| 66 | 1065 + 16528921 | 1158258 2156494837 × 1685724 1544984641 × 1953530 7537073043 × 2621729 3034303991 | Donovan Johnson (14 Sep 2007) |
| 67 | 1066 + 1037911 | 2129165 1498132551 × 2713341 3015756133 × 3393022 9792704437 × 5101518 0136202441 | Donovan Johnson (14 Sep 2007) |
| 68 | 1067 + 3136153 | 2233795 4099825591 × 7547860 3724282501 × 7657282 1197688237 × 7745654 1233638159 | Donovan Johnson (17 Sep 2007) |
| 69 | (1017 + 3)4 | 10000000 0000000003 × 10000000 0000000003 × 10000000 0000000003 × 10000000 0000000003 | Trivial |
| 70 | 1069 + 2495233 | 11105357 8017928633 × 11225151 3276295871 × 15080064 2421161699 × 53195151 2758103069 | Donovan Johnson (14 Sep 2007) |
| 71 | 1070 + 15754711 | 14581380 8529168361 × 20563178 0744229349 × 38966732 5576323679 × 85588835 8348768781 | Donovan Johnson (17 Sep 2007) |
| 73 | (1018 + 3)4 | 100000000 0000000003 × 100000000 0000000003 × 100000000 0000000003 × 100000000 0000000003 | Trivial |
| 75 | 1074 + 5773243 | 137595665 3530253917 × 277293315 6399582559 × 296346535 4433379559 × 884414781 2502586159 | Donovan Johnson (26 Sep 2007) |
The next table contains the maximal numbers with given number of figures whose four prime factors have the same length when expressed in decimal notation.
| Digits | Maximal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 90 | 2 × 3 × 3 × 5 | Bouk de Water |
| 3 | 875 | 5 × 5 × 5 × 7 | Bouk de Water |
| 4 | 2401 | 7 × 7 × 7 × 7 | Trivial |
| 5 | 99671 | 11 × 13 × 17 × 41 | Bouk de Water |
| 6 | 999973 | 13 × 13 × 61 × 97 | Bouk de Water |
| 7 | 107 − 8709 | 17 × 73 × 83 × 97 | Bouk de Water |
| 8 | 974 | 97 × 97 × 97 × 97 | Trivial |
| 9 | 109 − 2717 | 107 × 109 × 179 × 479 | Bouk de Water |
| 10 | 1010 − 1671 | 163 × 241 × 277 × 919 | Bouk de Water |
| 11 | 1011 − 2289 | 311 × 397 × 829 × 977 | Richard Heylen (16 May 2002) |
| 12 | 9974 | 997 × 997 × 997 × 997 | Trivial |
| 13 | 1013 − 191 | 1051 × 1117 × 2063 × 4129 | Bouk de Water |
| 14 | 1014 − 11021 | 2161 × 2297 × 3019 × 6673 | Bouk de Water |
| 15 | 1015 − 8459 | 2939 × 4957 × 7907 × 8681 | Bouk de Water |
| 16 | 99734 | 9973 × 9973 × 9973 × 9973 | Trivial |
| 17 | 1017 − 194271 | 13337 × 14537 × 17401 × 29641 | Bouk de Water |
| 18 | 1018 − 31247 | 15773 × 19301 × 40697 × 80713 | Richard Heylen (16 May 2002) |
| 19 | 1019 − 165507 | 31891 × 38453 × 83617 × 97523 | Richard Heylen (16 May 2002) |
| 20 | 999914 | 99991 × 99991 × 99991 × 99991 | Trivial |
| 21 | 1021 − 40167 | 107453 × 167747 × 179497 × 309079 | Richard Heylen (16 May 2002) |
| 22 | 1022 − 43197 | 171761 × 296827 × 439493 × 446293 | Richard Heylen (16 May 2002) |
| 23 | 1023 − 121227 | 393629 × 519031 × 625909 × 782003 | Richard Heylen (16 May 2002) |
| 24 | 9999834 | 999983 × 999983 × 999983 × 999983 | Trivial |
| 25 | 1025 − 225371 | 1090409 × 1356721 × 1794679 × 3766459 | Richard Heylen (16 May 2002) |
| 26 | 1026 − 12227 | 1494659 × 2763377 × 4873753 × 4967687 | Richard Heylen (16 May 2002) |
| 27 | 1027 − 181239 | 4668211 × 5065199 × 5361887 × 7887427 | Richard Heylen (22 May 2002) |
| 28 | (107 − 9)4 | 9999991 × 9999991 × 9999991 × 9999991 | Trivial |
| 29 | 1029 − 121587 | 10211363 × 14281931 × 22793899 × 30082279 | Sander Hoogendoorn |
| 30 | 1030 − 225131 | 14671121 × 16358479 × 60823691 × 68504801 | Sander Hoogendoorn |
| 31 | 1031 − 64409 | 43520599 × 50243803 × 63662927 × 71834989 | Sander Hoogendoorn |
| 32 | (108 − 11)4 | 99999989 × 99999989 × 99999989 × 99999989 | Trivial |
| 33 | 1033 − 2981361 | 138811943 × 150388417 × 172824791 × 277174159 | Richard Heylen (9 Aug 2002) |
| 34 | 1034 − 291003 | 160575161 × 169865251 × 543834983 × 674139769 | Richard Heylen (9 Aug 2002) |
| 35 | 1035 − 7133483 | 249412637 × 577688777 × 800184277 × 867356429 | Richard Heylen (9 Aug 2002) |
| 36 | (109 − 63)4 | 999999937 × 999999937 × 999999937 × 999999937 | Trivial |
| 37 | 1037 − 3392961 | 1024919407 × 1205624261 × 2704285817 × 2992579621 | Jim Fougeron |
| 38 | 1038 − 1209213 | 1277941309 × 3496358689 × 4329118921 × 5169798247 | Jim Fougeron |
| 39 | 1039 − 716501 | 2188752617 × 6832015219 × 7332337547 × 9120361379 | Jim Fougeron |
| 40 | (1010 − 33)4 | 9999999967 × 9999999967 × 9999999967 × 9999999967 | Trivial |
| 41 | 1041 − 1752149 | 1 1665636553 × 1 2020765051 × 1 2486083933 × 5 7112766749 | Jim Fougeron |
| 42 | 1042 − 1968921 | 1 6387356763 × 2 9844439783 × 3 5069159969 × 5 8304529779 | Jim Fougeron |
| 43 | 1043 − 812357 | 3 2266433293 × 6 6076452031 × 6 8130819217 × 6 8842829513 | Jim Fougeron |
| 44 | (1011 − 23)4 | 9 9999999977 × 9 9999999977 × 9 9999999977 × 9 9999999977 | Trivial |
| 45 | 1045 − 1021013 | 11 1193045207 × 16 2288480071 × 17 0369552917 × 32 5269018463 | Phil Carmody (26 Oct 2003) |
| 46 | 1046 − 550623 | 11 4257545709 × 30 1659452561 × 36 9064621643 × 78 6132506111 | Phil Carmody (26 Oct 2003) |
| 47 | 1047 − 12394547 | 25 8602566111 × 53 7704923703 × 78 4384962709 × 91 6840728049 | Phil Carmody (27 Oct 2003) |
| 48 | (1012 − 11)4 | 99 9999999989 × 99 9999999989 × 99 9999999989 × 99 9999999989 | Trivial |
| 49 | 1049 − 1328811 | 112 3363694533 × 113 2900299593 × 161 9376154261 × 485 2216658021 | Predrag Minovic (25 Apr 2006) |
| 50 | 1050 − 1827771 | 113 1830822071 × 229 9071133453 × 462 6382993013 × 830 6622844691 | Andreas Höglund (22 Aug 2006) |
| 51 | 1051 − 2839931 | 261 0618693979 × 498 1318597973 × 860 7852782717 × 893 3412874271 | Andreas Höglund (16 Sep 2006) |
| 52 | (1013 − 29)4 | 999 9999999971 × 999 9999999971 × 999 9999999971 × 999 9999999971 | Trivial |
| 53 | 1053 − 853581 | 1380 1633011217 × 1510 2457237171 × 1535 9336773961 × 3123 5570830697 | Richard Heylen |
| 54 | 1054 − 5526581 | 1503 9932275681 × 2844 1324348441 × 4448 4945266719 × 5255 2241384381 | Donovan Johnson (26 Sep 2007) |
| 55 | 1055 − 1372949 | 1999 3325988847 × 7348 5361492133 × 8095 2155039507 × 8407 8644846843 | Donovan Johnson (26 Sep 2007) |
| 56 | (1014 − 27)4 | 9999 9999999973 × 9999 9999999973 × 9999 9999999973 × 9999 9999999973 | Trivial |
| 57 | 1057 − 4541 | 10308 6207782083 × 15217 9523379973 × 21127 6757911589 × 30171 1242208009 | Richard Heylen (2 Jun 2003) |
| 58 | 1058 − 4187751 | 14342 8321659551 × 23611 9351144679 × 52398 1514063681 × 56353 0643461601 | Donovan Johnson (26 Sep 2007) |
| 59 | 1059 − 14655741 | 31373 6165207561 × 53613 5329528819 × 67203 5000546687 × 88464 5142291223 | Donovan Johnson (26 Sep 2007) |
| 60 | (1015 − 11)4 | 99999 9999999989 × 99999 9999999989 × 99999 9999999989 × 99999 9999999989 | Trivial |
| 61 | 1061 − 22336827 | 115611 2937602863 × 182156 8051966979 × 198522 8621423119 × 239190 4358614871 | Donovan Johnson (30 Sep 2007) |
| 62 | 1062 − 4626803 | 162259 9674057919 × 217200 8813516923 × 307388 4742899533 × 923080 2373690957 | Richard Heylen (17 Sep 2003) |
| 63 | 1063 − 34575809 | 323756 4338664067 × 341522 9830176679 × 944546 8489873021 × 957498 6131281247 | Donovan Johnson (9 Oct 2007) |
| 64 | (1016 − 63)4 | 999999 9999999937 × 999999 9999999937 × 999999 9999999937 × 999999 9999999937 | Trivial |
| 65 | 1065 − 11058639 | 1199199 9226304927 × 1419138 5939851301 × 1785036 8215222619 × 3291822 5659149497 | Donovan Johnson (3 Oct 2007) |
| 66 | 1066 − 1194869 | 1668960 8733528161 × 1887779 1141985067 × 3547100 7677518727 × 8948065 2326297719 | Donovan Johnson (30 Sep 2007) |
| 67 | 1067 − 139852557 | 3143197 4044879763 × 4612814 3017681421 × 7012542 4881114271 × 9835283 1098894971 | Alfred Reich (14 Feb 2021) |
| 68 | (1017 − 3)4 | 9999999 9999999997 × 9999999 9999999997 × 9999999 9999999997 × 9999999 9999999997 | Trivial |
| 69 | 1069 − 5603969 | 14678294 5262074273 × 15587349 6933692249 × 19412356 1207348719 × 22515099 6630849337 | Eric Jeancolas (19 Apr 2021) |
| 70 | 1070 − 15509327 | 12404739 2935238951 × 23524713 7707008731 × 56966178 0235940563 × 60154888 7561551591 | Eric Jeancolas (23 Mar 2021) |
| 72 | (1018 − 11)4 | 99999999 9999999989 × 99999999 9999999989 × 99999999 9999999989 × 99999999 9999999989 | Trivial |
If you have a number that can be in these tables or you have any comment please fill the form.
Written by Dario Alpern. Last updated on September 6th, 2021.