6-Brilliant numbers
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- 6-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 and greatest brilliant numbers with a given 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
Maximal and minimal base-2 2-brilliant numbers
Density of brilliant numbers (by Richard Heylen)
Minimal 6-brilliant numbers
The next table contains the minimal numbers with given number of figures whose six prime factors have the same length when expressed in decimal notation.
| Digits | Minimal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 64 | 2 × 2 × 2 × 2 × 2 × 2 | Andrea Fanchini (26 Feb 2025) |
| 3 | 144 | 2 × 2 × 2 × 2 × 3 × 3 | Andrea Fanchini (26 Feb 2025) |
| 4 | 1000 | 2 × 2 × 2 × 5 × 5 × 5 | Andrea Fanchini (26 Feb 2025) |
| 5 | 10290 | 2 × 3 × 5 × 7 × 7 × 7 | Andrea Fanchini (26 Feb 2025) |
| 6 | 117649 | 7 × 7 × 7 × 7 × 7 × 7 | Andrea Fanchini (26 Feb 2025) |
| 7 | 116 | 11 × 11 × 11 × 11 × 11 × 11 | Trivial |
| 8 | 107 + 81357 | 11 × 11 × 13 × 13 × 17 × 29 | Andrea Fanchini (26 Feb 2025) |
| 9 | 108 + 16059 | 11 × 11 × 13 × 13 × 67 × 73 | Andrea Fanchini (26 Feb 2025) |
| 10 | 109 + 52443 | 13 × 23 × 29 × 29 × 41 × 97 | Andrea Fanchini (26 Feb 2025) |
| 11 | 1010 + 108889 | 23 × 29 × 59 × 59 × 59 × 73 | Andrea Fanchini (26 Feb 2025) |
| 12 | 1011 + 197487 | 41 × 53 × 67 × 73 × 97 × 97 | Andrea Fanchini (26 Feb 2025) |
| 13 | 1016 | 101 × 101 × 101 × 101 × 101 × 101 | Trivial |
| 14 | 1013 + 1342381 | 101 × 101 × 149 × 163 × 181 × 223 | Andrea Fanchini (26 Feb 2025) |
| 15 | 1014 + 1274419 | 101 × 101 × 151 × 211 × 313 × 983 | Andrea Fanchini (26 Feb 2025) |
| 16 | 1015 + 407943 | 151 × 311 × 317 × 379 × 421 × 421 | Andrea Fanchini (26 Feb 2025) |
| 17 | 1016 + 2328283 | 151 × 397 × 467 × 599 × 691 × 863 | Andrea Fanchini (26 Feb 2025) |
| 18 | 1017 + 103742353 | 563 × 577 × 673 × 751 × 751 × 811 | Andrea Fanchini (26 Feb 2025) |
| 19 | 10096 | 1009 × 1009 × 1009 × 1009 × 1009 × 1009 | Trivial |
| 20 | 1019 + 150988593 | 1063 × 1069 × 1217 × 1399 × 1429 × 3617 | Andrea Fanchini (26 Feb 2025) |
| 21 | 1020 + 15252987 | 1021 × 1459 × 1543 × 2293 × 2789 × 6803 | Andrea Fanchini (26 Feb 2025) |
| 22 | 1021 + 1915071 | 1901 × 2777 × 3083 × 3581 × 3881 × 4421 | Andrea Fanchini (26 Feb 2025) |
| 23 | 1022 + 4579353 | 3137 × 3313 × 4481 × 4639 × 4877 × 9491 | Andrea Fanchini (26 Feb 2025) |
| 24 | 1023 + 1506039807 | 5431 × 5557 × 6679 × 6763 × 7853 × 9341 | Andrea Fanchini (26 Feb 2025) |
| 25 | 100076 | 10007 × 10007 × 10007 × 10007 × 10007 × 10007 | Trivial |
| 26 | 1025 + 1658826727 | 10301 × 10463 × 11743 × 11779 × 18521 × 36217 | Andrea Fanchini (26 Feb 2025) |
| 27 | 1026 + 27126793 | 10529 × 14057 × 15649 × 29983 × 32587 × 44189 | Andrea Fanchini (26 Feb 2025) |
| 28 | 1027 + 22402237 | 10459 × 10883 × 33503 × 42467 × 64717 × 95413 | Andrea Fanchini (26 Feb 2025) |
| 29 | 1028 + 772267 | 19963 × 26431 × 55619 × 58441 × 69427 × 83983 | Andrea Fanchini (26 Feb 2025) |
| 30 | 1029 + 129203833 | 44257 × 51239 × 76771 × 79273 × 84127 × 86131 | Andrea Fanchini (26 Feb 2025) |
| 31 | 1000036 | 100003 × 100003 × 100003 × 100003 × 100003 × 100003 | Trivial |
| 32 | 1031 + 8378421841 | 106781 × 127973 × 131297 × 132439 × 159791 × 263369 | Eric Jeancolas (17 Apr 2025) |
| 33 | 1032 + 164120001 | 126781 × 135601 × 138251 × 225431 × 262543 × 710887 | Andrea Fanchini (26 Feb 2025) |
| 34 | 1033 + 2794567 | 118033 × 169639 × 295081 × 393919 × 441877 × 972347 | Andrea Fanchini (26 Feb 2025) |
| 35 | 1034 + 373735401 | 129281 × 459463 × 573379 × 628013 × 649501 × 719821 | Andrea Fanchini (26 Feb 2025) |
| 36 | 1035 + 329846779 | 444637 × 560281 × 685039 × 728873 × 818621 × 982061 | Andrea Fanchini (26 Feb 2025) |
| 37 | (106 + 3)6 | 1000003 × 1000003 × 1000003 × 1000003 × 1000003 × 1000003 | Trivial |
| 38 | 1037 + 2547895147 | 1000847 × 1150847 × 1213747 × 1598263 × 2062483 × 2169941 | Eric Jeancolas (22 Apr 2025) |
| 39 | 1038 + 249049843 | 1235929 × 1816091 × 1877243 × 2298103 × 3086159 × 3346267 | Andrea Fanchini (26 Feb 2025) |
| 40 | 1039 + 155896629 | 1047283 × 2105431 × 3114971 × 3648289 × 5134117 × 7772951 | Andrea Fanchini (26 Feb 2025) |
| 41 | 1040 + 1155626023 | 2104933 × 3436357 × 4609079 × 5417567 × 6984167 × 7927393 | Eric Jeancolas (7 May 2025) |
| 42 | 1041 + 4 0010207353 | 5060857 × 5618027 × 6093881 × 7291597 × 8575037 × 9230803 | Eric Jeancolas (11 Sep 2025) |
| 43 | (107 + 19)6 | 10000019 × 10000019 × 10000019 × 10000019 × 10000019 × 10000019 | Trivial |
| 45 | 1044 + 1363540563 | 11824781 × 12400097 × 13283357 × 18029051 × 49610527 × 57402031 | Eric Jeancolas (29 Aug 2025) |
| 46 | 1045 + 2569474239 | 11371693 × 21388649 × 27929507 × 30905531 × 51292537 × 92861963 | Eric Jeancolas (10 Sep 2025) |
Maximal 6-brilliant numbers
The next table contains the maximal numbers with given number of figures whose six prime factors have the same length when expressed in decimal notation.
| Digits | Maximal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 96 | 2 × 2 × 2 × 2 × 2 × 3 | Andrea Fanchini (26 Feb 2025) |
| 3 | 900 | 2 × 2 × 3 × 3 × 5 × 5 | Dario Alpern (26 Feb 2025) |
| 4 | 9604 | 2 × 2 × 7 × 7 × 7 × 7 | Andrea Fanchini (26 Feb 2025) |
| 5 | 84035 | 5 × 7 × 7 × 7 × 7 × 7 | Andrea Fanchini (26 Feb 2025) |
| 6 | 117649 | 7 × 7 × 7 × 7 × 7 × 7 | Trivial |
| 7 | 107 − 112001 | 11 × 11 × 11 × 17 × 19 × 23 | Andrea Fanchini (26 Feb 2025) |
| 8 | 108 − 9593 | 11 × 11 × 19 × 23 × 31 × 61 | Andrea Fanchini (26 Feb 2025) |
| 9 | 109 − 21449 | 11 × 13 × 29 × 59 × 61 × 67 | Andrea Fanchini (26 Feb 2025) |
| 10 | 1010 − 115263 | 19 × 31 × 43 × 67 × 71 × 83 | Andrea Fanchini (26 Feb 2025) |
| 11 | 1011 − 1886931 | 47 × 61 × 61 × 83 × 83 × 83 | Andrea Fanchini (26 Feb 2025) |
| 12 | 976 | 97 × 97 × 97 × 97 × 97 × 97 | Trivial |
| 13 | 1013 − 34479491 | 101 × 107 × 113 × 197 × 197 × 211 | Andrea Fanchini (26 Feb 2025) |
| 14 | 1014 − 746861 | 113 × 181 × 199 × 199 × 331 × 373 | Andrea Fanchini (26 Feb 2025) |
| 15 | 1015 − 1433997 | 131 × 167 × 191 × 379 × 653 × 967 | Andrea Fanchini (26 Feb 2025) |
| 16 | 1016 − 1821479 | 131 × 353 × 467 × 647 × 751 × 953 | Andrea Fanchini (26 Feb 2025) |
| 17 | 1017 − 113998721 | 359 × 631 × 673 × 821 × 877 × 911 | Andrea Fanchini (26 Feb 2025) |
| 18 | 9976 | 997 × 997 × 997 × 997 × 997 × 997 | Trivial |
| 19 | 1019 − 50525853 | 1019 × 1093 × 1283 × 1297 × 1693 × 3187 | Andrea Fanchini (26 Feb 2025) |
| 20 | 1020 − 363479 | 1051 × 1229 × 1987 × 2423 × 3209 × 5011 | Andrea Fanchini (26 Feb 2025) |
| 21 | 1021 − 17104089 | 1439 × 1583 × 2549 × 2927 × 7417 × 7933 | Andrea Fanchini (26 Feb 2025) |
| 22 | 1022 − 1447209 | 1847 × 2087 × 5557 × 6037 × 7937 × 9743 | Andrea Fanchini (26 Feb 2025) |
| 23 | 1023 − 2382283763 | 5479 × 5581 × 6143 × 6907 × 8563 × 9001 | Andrea Fanchini (26 Feb 2025) |
| 24 | 99736 | 9973 × 9973 × 9973 × 9973 × 9973 × 9973 | Trivial |
| 25 | 1025 − 763857411 | 11149 × 12421 × 14149 × 16067 × 17483 × 18169 | Andrea Fanchini (26 Feb 2025) |
| 26 | 1026 − 25284551 | 10663 × 15227 × 17077 × 26713 × 36469 × 37021 | Andrea Fanchini (26 Feb 2025) |
| 27 | 1027 − 50949951 | 11299 × 12343 × 22349 × 58613 × 73943 × 74027 | Andrea Fanchini (26 Feb 2025) |
| 28 | 1028 − 17324201 | 33871 × 41947 × 42643 × 46771 × 56533 × 62423 | Andrea Fanchini (26 Feb 2025) |
| 29 | 1029 − 1753331019 | 45403 × 59629 × 64439 × 77813 × 84731 × 86939 | Andrea Fanchini (26 Feb 2025) |
| 30 | 999916 | 99991 × 99991 × 99991 × 99991 × 99991 × 99991 | Trivial |
| 31 | 1031 − 8160991449 | 100333 × 131441 × 136069 × 147629 × 185557 × 203431 | Alfred Reich (28 Mar 2025) |
| 32 | 1032 − 640007759 | 103651 × 136999 × 187069 × 211039 × 214481 × 831679 | Andrea Fanchini (26 Feb 2025) |
| 33 | 1033 − 6588249 | 172297 × 236893 × 258539 × 364541 × 444929 × 584261 | Andrea Fanchini (26 Feb 2025) |
| 34 | 1034 − 815987867 | 141863 × 333679 × 575669 × 588191 × 760927 × 819913 | Alfred Reich (29 Mar 2025) |
| 35 | 1035 − 14972647671 | 307651 × 580733 × 839539 × 860369 × 864221 × 896633 | Alfred Reich (30 Mar 2025) |
| 36 | (106 − 17)6 | 999983 × 999983 × 999983 × 999983 × 999983 × 999983 | Trivial |
| 37 | 1037 − 20205420381 | 1149349 × 1210207 × 1300283 × 1588729 × 1711397 × 2033527 | Alfred Reich (24 Apr 2025) |
| 38 | 1038 − 262866569 | 1212931 × 1221793 × 1381637 × 1559891 × 3475897 × 9007643 | Eric Jeancolas (20 Apr 2025) |
| 39 | 1039 − 63766727 | 1173127 × 2089523 × 2092501 × 2978033 × 7002227 × 9349243 | Eric Jeancolas (26 Mar 2025) |
| 40 | 1040 − 111370113 | 1890397 × 3032801 × 5472541 × 5486599 × 6210041 × 9354409 | Eric Jeancolas (20 Apr 2025) |
| 41 | 1041 − 1 0072828733 | 3131263 × 6080677 × 6236719 × 8927579 × 9599423 × 9826379 | Alfred Reich (12 Jul 2025) |
| 42 | (107 − 9)6 | 9999991 × 9999991 × 9999991 × 9999991 × 9999991 × 9999991 | Trivial |
| 44 | 1044 − 254652041 | 13062611 × 17642657 × 22103087 × 22228883 × 22890377 × 38581801 | Alfred Reich (26 Mar 2025) |
| 45 | 1045 − 498426503 | 18188749 × 26201569 × 27434843 × 40434209 × 42856207 × 44137193 | Eric Jeancolas (25 Aug 2025) |
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 .