5-Brilliant numbers
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- 5-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 5-brilliant numbers
The next table contains the minimal numbers with given number of figures whose five prime factors have the same length when expressed in decimal notation.
| Digits | Minimal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 32 | 2 × 2 × 2 × 2 × 2 | Andrea Fanchini (3 Jan 2025) |
| 3 | 108 | 2 × 2 × 3 × 3 × 3 | Andrea Fanchini (3 Jan 2025) |
| 4 | 1050 | 2 × 3 × 5 × 5 × 7 | Andrea Fanchini (3 Jan 2025) |
| 5 | 12005 | 5 × 7 × 7 × 7 × 7 | Andrea Fanchini (3 Jan 2025) |
| 6 | 161051 | 11 × 11 × 11 × 11 × 11 | Trivial |
| 7 | 106 + 4663 | 11 × 11 × 19 × 19 × 23 | Andrea Fanchini (3 Jan 2025) |
| 8 | 107 + 331 | 11 × 23 × 29 × 29 × 47 | Andrea Fanchini (3 Jan 2025) |
| 9 | 108 + 13227 | 17 × 41 × 43 × 47 × 71 | Andrea Fanchini (3 Jan 2025) |
| 10 | 109 + 101527 | 37 × 61 × 71 × 79 × 79 | Andrea Fanchini (3 Jan 2025) |
| 11 | 1015 | 101 × 101 × 101 × 101 × 101 | Trivial |
| 12 | 1011 + 57123 | 103 × 157 × 179 × 179 × 193 | Andrea Fanchini (3 Jan 2025) |
| 13 | 1012 + 10839 | 131 × 167 × 197 × 331 × 701 | Andrea Fanchini (3 Jan 2025) |
| 14 | 1013 + 101881 | 173 × 257 × 433 × 521 × 997 | Andrea Fanchini (3 Jan 2025) |
| 15 | 1014 + 119463 | 337 × 557 × 757 × 797 × 883 | Andrea Fanchini (3 Jan 2025) |
| 16 | 10095 | 1009 × 1009 × 1009 × 1009 × 1009 | Trivial |
| 17 | 1016 + 1178601 | 1061 × 1423 × 1607 × 1831 × 2251 | Andrea Fanchini |
| 18 | 1017 + 154711 | 1319 × 1741 × 2309 × 2521 × 7481 | Andrea Fanchini (3 Jan 2025) |
| 19 | 1018 + 463929 | 1607 × 3583 × 4099 × 4673 × 9067 | Andrea Fanchini (3 Jan 2025) |
| 20 | 1019 + 1818523 | 2539 × 6311 × 8263 × 8419 × 8971 | Andrea Fanchini (3 Jan 2025) |
| 21 | 100075 | 10007 × 10007 × 10007 × 10007 × 10007 | Trivial |
| 22 | 1021 + 2197711 | 10139 × 12161 × 16651 × 18121 × 26879 | Andrea Fanchini (3 Jan 2025) |
| 23 | 1022 + 858997 | 11701 × 13883 × 28439 × 39847 × 54323 | Andrea Fanchini (3 Jan 2025) |
| 24 | 1023 + 508831 | 18913 × 35509 × 46381 × 50231 × 63913 | Andrea Fanchini (3 Jan 2025) |
| 25 | 1024 + 12872143 | 29879 × 45893 × 85381 × 87911 × 97159 | Andrea Fanchini (3 Jan 2025) |
| 26 | (105 + 3)5 | 100003 × 100003 × 100003 × 100003 × 100003 | Trivial |
| 27 | 1026 + 2254263 | 100847 × 157649 × 159589 × 188311 × 209299 | Andrea Fanchini (3 Jan 2025) |
| 28 | 1027 + 4674427 | 112687 × 146833 × 341491 × 418349 × 423043 | Andrea Fanchini (3 Jan 2025) |
| 29 | 1028 + 1296663 | 157219 × 373003 × 420691 × 542131 × 747679 | Andrea Fanchini (3 Jan 2025) |
| 30 | 1029 + 28986933 | 341749 × 383987 × 897433 × 918949 × 924023 | Andrea Fanchini (3 Jan 2025) |
| 31 | (106 + 3)5 | 1000003 × 1000003 × 1000003 × 1000003 × 1000003 | Trivial |
| 32 | 1031 + 70584643 | 1025261 × 1313087 × 1841339 × 1892309 × 2131799 | Andrea Fanchini (23 Jan 2025) |
| 33 | 1032 + 4706367 | 1732727 × 1932877 × 2041489 × 2744029 × 5330033 | Andrea Fanchini (23 Jan 2025) |
| 34 | 1033 + 16562067 | 1694447 × 2422327 × 4675859 × 5980747 × 8712091 | Andrea Fanchini (23 Jan 2025) |
| 35 | 1034 + 32605507 | 4297519 × 5389313 × 5909671 × 7716103 × 9468637 | Andrea Fanchini (23 Jan 2025) |
| 36 | (107 + 19)5 | 10000019 × 10000019 × 10000019 × 10000019 × 10000019 | Trivial |
| 37 | 1036 + 9335739 | 10962779 × 13862341 × 15493253 × 20545939 × 20671603 | Andrea Fanchini (23 Jan 2025) |
| 38 | 1037 + 201729 | 10058089 × 17668139 × 32225341 × 37006811 × 47186149 | Andrea Fanchini (23 Jan 2025) |
| 39 | 1038 + 27607063 | 14018273 × 19966897 × 40926371 × 90002893 × 96991841 | Andrea Fanchini (23 Jan 2025) |
| 40 | 1039 + 14955157 | 34959251 × 44811743 × 71847299 × 92388281 × 96165371 | Andrea Fanchini (23 Jan 2025) |
| 41 | (108 + 7)5 | 100000007 × 100000007 × 100000007 × 100000007 × 100000007 | Trivial |
| 42 | 1041 + 318101823 | 102765517 × 103880737 × 152779691 × 213778463 × 286805839 | Eric Jeancolas (22 Apr 2025) |
| 43 | 1042 + 9565851 | 111487069 × 255807911 × 296075891 × 317616317 × 372868487 | Eric Jeancolas (26 Mar 2025) |
| 44 | 1043 + 47640291 | 116622971 × 211666387 × 618833113 × 708018769 × 924582539 | Eric Jeancolas (8 Mar 2025) |
| 45 | 1044 + 873830527 | 175008643 × 791151997 × 852523757 × 890374451 × 951483791 | Eric Jeancolas (16 Jul 2025) |
| 46 | (109 + 7)5 | 1000000007 × 1000000007 × 1000000007 × 1000000007 × 1000000007 | Trivial |
| 47 | 1046 + 1339779687 | 1296410657 × 1400895877 × 1567941797 × 1855041233 × 1893075983 | Eric Jeancolas (4 May 2026) |
| 48 | 1047 + 23558053 | 1253154601 × 1664433697 × 1679585333 × 3209172311 × 8894749823 | Eric Jeancolas (1 Apr 2025) |
| 49 | 1048 + 4940391 | 1288629179 × 1881936571 × 6688556099 × 7763112367 × 7941431803 | Eric Jeancolas (6 Apr 2025) |
| 50 | 1049 + 407612557 | 3190217609 × 4633825459 × 7410254191 × 9193643951 × 9929313767 | Eric Jeancolas (2 May 2026) |
| 51 | (1010 + 19)5 | 1 0000000019 × 1 0000000019 × 1 0000000019 × 1 0000000019 × 1 0000000019 | Trivial |
| 52 | 1051 + 129556849 | 1 0778991187 × 1 3117895911 × 1 3434620659 × 1 6596009359 × 3 1719667297 | Eric Jeancolas (2 May 2026) |
| 53 | 1052 + 66667411 | 1 0951712633 × 1 7247606881 × 2 6339562853 × 3 3164077369 × 6 0605582951 | Eric Jeancolas (15 Apr 2025) |
| 54 | 1053 + 70098819 | 1 9163861539 × 3 1165252583 × 3 6820919203 × 5 9186320477 × 7 6829915777 | Eric Jeancolas (11 Apr 2025) |
Maximal 5-brilliant numbers
The next table contains the maximal numbers with given number of figures whose five prime factors have the same length when expressed in decimal notation.
| Digits | Maximal | Prime factors | Discoverer |
|---|---|---|---|
| 2 | 80 | 2 × 2 × 2 × 2 × 5 | Andrea Fanchini (25 Jan 2025) |
| 3 | 980 | 2 × 2 × 5 × 7 × 7 | Andrea Fanchini (25 Jan 2025) |
| 4 | 8575 | 5 × 5 × 7 × 7 × 7 | Andrea Fanchini (25 Jan 2025) |
| 5 | 16807 | 7 × 7 × 7 × 7 × 7 | Trivial |
| 6 | 990847 | 11 × 13 × 13 × 13 × 41 | Andrea Fanchini (25 Jan 2025) |
| 7 | 107 − 5411 | 11 × 17 × 19 × 29 × 97 | Alfred Reich (22 Jan 2021) |
| 8 | 108 − 4317 | 17 × 29 × 43 × 53 × 89 | Alfred Reich (22 Jan 2021) |
| 9 | 109 − 797979 | 41 × 61 × 67 × 67 × 89 | Alfred Reich (22 Jan 2021) |
| 10 | 975 | 97 × 97 × 97 × 97 × 97 | Trivial |
| 11 | 1011 − 423887 | 101 × 103 × 113 × 257 × 331 | Alfred Reich (22 Jan 2021) |
| 12 | 1012 − 59901 | 131 × 131 × 211 × 277 × 997 | Alfred Reich (22 Jan 2021) |
| 13 | 1013 − 50327 | 281 × 293 × 347 × 571 × 613 | Alfred Reich (22 Jan 2021) |
| 14 | 1014 − 495633 | 181 × 751 × 859 × 859 × 997 | Alfred Reich (22 Jan 2021) |
| 15 | 9975 | 997 × 997 × 997 × 997 × 997 | Trivial |
| 16 | 1016 − 1176773 | 1033 × 1229 × 1319 × 1447 × 4127 | Alfred Reich (22 Jan 2021) |
| 17 | 1017 − 416447 | 1327 × 1483 × 3001 × 3371 × 5023 | Alfred Reich (22 Jan 2021) |
| 18 | 1018 − 343229 | 1583 × 2707 × 3691 × 6577 × 9613 | Alfred Reich (22 Jan 2021) |
| 19 | 1019 − 6735117 | 3041 × 4129 × 8581 × 9433 × 9839 | Alfred Reich (22 Jan 2021) |
| 20 | 99735 | 9973 × 9973 × 9973 × 9973 × 9973 | Trivial |
| 21 | 1021 − 3964821 | 10007 × 13537 × 16811 × 16921 × 25951 | Alfred Reich (22 Jan 2021) |
| 22 | 1022 − 1218891 | 11807 × 15391 × 22189 × 35809 × 69257 | Alfred Reich (22 Jan 2021) |
| 23 | 1023 − 3227433 | 10253 × 45673 × 46573 × 61729 × 74279 | Alfred Reich (22 Jan 2021) |
| 24 | 1024 − 2962853 | 21961 × 65563 × 79309 × 92623 × 94547 | Alfred Reich (22 Jan 2021) |
| 25 | 999915 | 99991 × 99991 × 99991 × 99991 × 99991 | Trivial |
| 26 | 1026 − 13996773 | 104471 × 105361 × 150889 × 219017 × 274909 | Alfred Reich (22 Jan 2021) |
| 27 | 1027 − 2939079 | 107837 × 133669 × 157349 × 468113 × 941861 | Alfred Reich (22 Jan 2021) |
| 28 | 1028 − 575559 | 231269 × 236563 × 445649 × 486821 × 842507 | Alfred Reich (22 Jan 2021) |
| 29 | 1029 − 13773393 | 303871 × 689957 × 709909 × 747157 × 899237 | Alfred Reich (22 Jan 2021) |
| 30 | (106 − 17)5 | 999983 × 999983 × 999983 × 999983 × 999983 | Trivial |
| 31 | 1031 − 5846847 | 1022701 × 1176899 × 1384829 × 2319659 × 2586377 | Alfred Reich (22 Jan 2021) |
| 32 | 1032 − 6088257 | 1679213 × 1796947 × 2128781 × 2156351 × 7219523 | Alfred Reich (22 Jan 2021) |
| 33 | 1033 − 992687 | 1743487 × 3112091 × 5050681 × 5613833 × 6500093 | Alfred Reich (22 Jan 2021) |
| 34 | 1034 − 8846847 | 3448663 × 4321529 × 7868789 × 9038629 × 9434119 | Alfred Reich (22 Jan 2021) |
| 35 | (107 − 9)5 | 9999991 × 9999991 × 9999991 × 9999991 × 9999991 | Trivial |
| 36 | 1036 − 117140009 | 11983739 × 11997917 × 13976143 × 19075949 × 26087251 | Alfred Reich (22 Jan 2021) |
| 37 | 1037 − 3737967 | 12265069 × 12316999 × 26021071 × 28269503 × 89987411 | Alfred Reich (22 Jan 2021) |
| 38 | 1038 − 17945273 | 14859149 × 26589149 × 49138601 × 69387613 × 74232979 | Alfred Reich (22 Jan 2021) |
| 39 | 1039 − 170690097 | 47957491 × 57090881 × 62357359 × 72085207 × 81253661 | Alfred Reich (22 Jan 2021) |
| 40 | (108 − 11)5 | 99999989 × 99999989 × 99999989 × 99999989 × 99999989 | Trivial |
| 41 | 1041 − 423906303 | 103948841 × 114710363 × 149953891 × 203482813 × 274847773 | Alfred Reich (22 Jan 2021) |
| 42 | 1042 − 46553637 | 108481379 × 192785767 × 207688121 × 301662217 × 763198063 | Alfred Reich (24 Mar 2021) |
| 43 | 1043 − 15301041 | 104012933 × 175136021 × 748629481 × 781365433 × 938460631 | Alfred Reich (24 Mar 2021) |
| 44 | 1044 − 1561244849 | 182407289 × 762137381 × 766498163 × 960663391 × 976882183 | Alfred Reich (24 Mar 2021) |
| 45 | (109 − 63)5 | 999999937 × 999999937 × 999999937 × 999999937 × 999999937 | Trivial |
| 46 | 1046 − 360258269 | 1055755553 × 1528954747 × 1668451013 × 1794793691 × 2068778527 | Alfred Reich (15 Apr 2021) |
| 47 | 1047 − 4359863 | 1214472871 × 2147253617 × 2409141583 × 3616902953 × 4400780009 | Eric Jeancolas (30 Mar 2025) |
| 48 | 1048 − 19443339 | 1039529507 × 4173506713 × 4220264131 × 6326926399 × 8632362059 | Alfred Reich (15 Apr 2021) |
| 49 | 1049 − 292349261 | 4199063917 × 6046596823 × 6692964461 × 7518662791 × 7826676379 | Alfred Reich (15 Apr 2021) |
| 50 | (1010 − 33)5 | 9999999967 × 9999999967 × 9999999967 × 9999999967 × 9999999967 | Trivial |
| 51 | 1051 − 43842761 | 1 1051449567 × 1 1579103727 × 1 3759812061 × 1 8480328409 × 3 0731490379 | Alfred Reich (15 Apr 2021) |
| 52 | 1052 − 25070787 | 1 0475873197 × 1 8031829987 × 2 3105205853 × 2 7132896251 × 8 4443060789 | Alfred Reich (15 Apr 2021) |
| 53 | 1053 − 189643463 | 1 7070660073 × 3 2560543801 × 4 2510933119 × 5 7999734231 × 7 2967823921 | Eric Jeancolas (28 Apr 2025) |
| 67 | 1067 − 138658263 | 1100 6813789669 × 1249 4334619307 × 2936 2240177729 × 3255 7607222003 × 7606 4774402597 | Alfred Reich (26 Dec 2024) |
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 May 7th, 2026.