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# 4-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.

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 4-brilliant numbers

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.

Minimal 4-brilliant numbers
DigitsMinimalPrime factorsDiscoverer
2162x
2 x
2 x
2
Bouk de Water
31002 x
2 x
5 x
5
Bouk de Water
410293 x
7 x
7 x
7
Bouk de Water
51464111 x
11 x
11 x
11
Trivial
610052911 x
13 x
19 x
37
Bouk de Water
7106 + 10911 x
23 x
59 x
67
Bouk de Water
8107 + 564719 x
61 x
89 x
97
Richard Heylen
(16 May 2002)
91014101 x
101 x
101 x
101
Trivial
10109 + 2847107 x
167 x
191 x
293
Bouk de Water
111010 + 861103 x
157 x
653 x
947
Bouk de Water
121011 + 24219389 x
401 x
643 x
997
Bouk de Water
13100941009 x
1009 x
1009 x
1009
Trivial
141013 + 541291321 x
1423 x
2087 x
2549
Bouk de Water
151014 + 97291033 x
3109 x
5479 x
5683
Bouk de Water
161015 + 117993499 x
5741 x
6133 x
8117
Bouk de Water
1710007410007 x
10007 x
10007 x
10007
Trivial
181017 + 3524111717 x
13469 x
18047 x
35111
Richard Heylen
(16 May 2002)
191018 + 1286120327 x
26849 x
27431 x
66797
Richard Heylen
(16 May 2002)
201019 + 8010127103 x
63317 x
68213 x
85427
Richard Heylen
(16 May 2002)
211000034100003 x
100003 x
100003 x
100003
Trivial
221021 + 200239100057 x
123923 x
164429 x
490481
Richard Heylen
(16 May 2002)
231022 + 146433140477 x
272759 x
309157 x
844183
Richard Heylen
(16 May 2002)
241023 + 22597304553 x
460841 x
765503 x
930763
Richard Heylen
(16 May 2002)
25(106 + 3)41000003 x
1000003 x
1000003 x
1000003
Trivial
261025 + 7194071178231 x
1184987 x
1761671 x
4065661
Richard Heylen
(16 May 2002)
271026 + 440471157759 x
2399869 x
4320469 x
8330353
Richard Heylen
(16 May 2002)
281027 + 3581732410043 x
4761727 x
9033361 x
9646313
Richard Heylen
(22 May 2002)
29(107 + 19)410000019 x
10000019 x
10000019 x
10000019
Trivial
301029 + 78432913503449 x
15690811 x
21475043 x
21977377
Jim Fougeron
311030 + 4619724847783 x
26380859 x
37443787 x
40742123
Jim Fougeron
321031 + 12053140738811 x
52961729 x
56976001 x
81346249
Jim Fougeron
33(108 + 7)4100000007 x
100000007 x
100000007 x
100000007
Trivial
341033 + 6379121022911 x
132294061 x
169058819 x
369448771
Jim Fougeron
351034 + 1167691117930893 x
237943709 x
381975661 x
932958863
Jim Fougeron
361035 + 1553517229833959 x
596374529 x
837572501 x
871052447
Jim Fougeron
37(109 + 7)41000000007 x
1000000007 x
1000000007 x
1000000007
Trivial
381037 + 732691015004657 x
2113986011 x
2124717653 x
2193454499
Jim Fougeron
391038 + 15056192535959857 x
3100290539 x
3280439369 x
3877244737
Jim Fougeron
401039 + 55608792806287299 x
5684681111 x
7330103057 x
8551684723
Jim Fougeron
41(1010 + 19)41 0000000019 x
1 0000000019 x
1 0000000019 x
1 0000000019
Trivial
421041 + 2703871 0057723799 x
1 4695606843 x
1 6439060371 x
4 1156248421
Jim Fougeron
431042 + 1902811 2904941277 x
2 8529199667 x
5 1682064947 x
5 2555058197
Jim Fougeron
441043 + 1865413 6717661187 x
4 0254606487 x
7 1613321583 x
9 4474714183
Jim Fougeron
45(1011 + 3)410 0000000003 x
10 0000000003 x
10 0000000003 x
10 0000000003
Trivial
461045 + 594572710 9637540401 x
13 0495801231 x
15 7056159707 x
44 5029913331
Richard Heylen
(11 Jul 2002)
471046 + 394050311 8764538069 x
21 4272532333 x
51 3125859067 x
76 5813137317
Richard Heylen
(19 Jul 2002)
481047 + 626544125 5256467691 x
67 8739441577 x
70 5264620759 x
81 8404596557
Donovan Johnson
(30 Aug 2007)
49(1012 + 39)4100 0000000039 x
100 0000000039 x
100 0000000039 x
100 0000000039
Trivial
501049 + 679663115 9788556469 x
117 5245695677 x
197 6861376901 x
371 1216539651
Richard Heylen
(2 Aug 2002)
511050 + 874531203 1459510449 x
264 6219932437 x
382 1581909133 x
486 7687850339
Sander Hoogendoorn
(17 Nov 2003)
521051 + 24031807358 6045996369 x
515 5570211641 x
670 6659821857 x
806 4940359719
Andreas Höglund
(16 Sep 2006)
53(1013 + 37)41000 0000000037 x
1000 0000000037 x
1000 0000000037 x
1000 0000000037
Trivial
541053 + 26260771251 7623121403 x
1277 7073989481 x
1666 7362777847 x
3751 2830115337
Donovan Johnson
(30 Aug 2007)
551054 + 12936791573 9418315123 x
3137 1181661341 x
3817 3288431427 x
5305 4334421939
Donovan Johnson
(7 Sep 2007)
561055 + 260829313146 5199992697 x
5099 8056344693 x
6675 9528591367 x
9334 7484189833
Donovan Johnson
(3 Oct 2007)
57(1014 + 31)410000 0000000031 x
10000 0000000031 x
10000 0000000031 x
10000 0000000031
Trivial
581057 + 181125311785 6492897181 x
13191 7327092977 x
20453 2887090121 x
31447 1646669889
Richard Heylen
(3 Sep 2002)
591058 + 580314712226 2486564571 x
25837 2872730919 x
55590 5946893789 x
56945 3860946227
Donovan Johnson
(12 Sep 2007)
601059 + 3671158926654 2160903959 x
62224 5970511791 x
63009 9845622983 x
95689 1329848907
Donovan Johnson
(3 Oct 2007)
61(1015 + 37)4100000 0000000037 x
100000 0000000037 x
100000 0000000037 x
100000 0000000037
Trivial
621061 + 470193145651 3870035241 x
173148 2834726159 x
175238 9428378763 x
226274 9930928469
Donovan Johnson
(12 Sep 2007)
631062 + 5111959227914 3986837019 x
277991 2264720531 x
357665 8593158251 x
441285 5031576581
Donovan Johnson
(12 Sep 2007)
641063 + 18799597265961 8573195897 x
533940 3019173071 x
757013 4814101641 x
930217 1161690691
Donovan Johnson
(17 Sep 2007)
65(1016 + 61)41000000 0000000061 x
1000000 0000000061 x
1000000 0000000061 x
1000000 0000000061
Trivial
661065 + 165289211158258 2156494837 x
1685724 1544984641 x
1953530 7537073043 x
2621729 3034303991
Donovan Johnson
(14 Sep 2007)
671066 + 10379112129165 1498132551 x
2713341 3015756133 x
3393022 9792704437 x
5101518 0136202441
Donovan Johnson
(14 Sep 2007)
681067 + 31361532233795 4099825591 x
7547860 3724282501 x
7657282 1197688237 x
7745654 1233638159
Donovan Johnson
(17 Sep 2007)
69(1017 + 3)410000000 0000000003 x
10000000 0000000003 x
10000000 0000000003 x
10000000 0000000003
Trivial
701069 + 249523311105357 8017928633 x
11225151 3276295871 x
15080064 2421161699 x
53195151 2758103069
Donovan Johnson
(14 Sep 2007)
711070 + 1575471114581380 8529168361 x
20563178 0744229349 x
38966732 5576323679 x
85588835 8348768781
Donovan Johnson
(17 Sep 2007)
751074 + 5773243137595665 3530253917 x
277293315 6399582559 x
296346535 4433379559 x
884414781 2502586159
Donovan Johnson
(26 Sep 2007)

## Maximal 4-brilliant numbers

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.

Maximal 4-brilliant numbers
DigitsMaximalPrime factorsDiscoverer
2902 x
3 x
3 x
5
Bouk de Water
38755 x
5 x
5 x
7
Bouk de Water
424017 x
7 x
7 x
7
Trivial
59967111 x
13 x
17 x
41
Bouk de Water
699997313 x
13 x
61 x
97
Bouk de Water
7107 - 870917 x
73 x
83 x
97
Bouk de Water
897497 x
97 x
97 x
97
Trivial
9109 - 2717107 x
109 x
179 x
479
Bouk de Water
101010 - 1671163 x
241 x
277 x
919
Bouk de Water
111011 - 2289311 x
397 x
829 x
977
Richard Heylen
(16 May 2002)
129974997 x
997 x
997 x
997
Trivial
131013 - 1911051 x
1117 x
2063 x
4129
Bouk de Water
141014 - 110212161 x
2297 x
3019 x
6673
Bouk de Water
151015 - 84592939 x
4957 x
7907 x
8681
Bouk de Water
16997349973 x
9973 x
9973 x
9973
Trivial
171017 - 19427113337 x
14537 x
17401 x
29641
Bouk de Water
181018 - 3124715773 x
19301 x
40697 x
80713
Richard Heylen
(16 May 2002)
191019 - 16550731891 x
38453 x
83617 x
97523
Richard Heylen
(16 May 2002)
2099991499991 x
99991 x
99991 x
99991
Trivial
211021 - 40167107453 x
167747 x
179497 x
309079
Richard Heylen
(16 May 2002)
221022 - 43197171761 x
296827 x
439493 x
446293
Richard Heylen
(16 May 2002)
231023 - 121227393629 x
519031 x
625909 x
782003
Richard Heylen
(16 May 2002)
249999834999983 x
999983 x
999983 x
999983
Trivial
251025 - 2253711090409 x
1356721 x
1794679 x
3766459
Richard Heylen
(16 May 2002)
261026 - 122271494659 x
2763377 x
4873753 x
4967687
Richard Heylen
(16 May 2002)
271027 - 1812394668211 x
5065199 x
5361887 x
7887427
Richard Heylen
(22 May 2002)
28(107 - 9)49999991 x
9999991 x
9999991 x
9999991
Trivial
291029 - 12158710211363 x
14281931 x
22793899 x
30082279
Sander Hoogendoorn
301030 - 22513114671121 x
16358479 x
60823691 x
68504801
Sander Hoogendoorn
311031 - 6440943520599 x
50243803 x
63662927 x
71834989
Sander Hoogendoorn
32(108 - 11)499999989 x
99999989 x
99999989 x
99999989
Trivial
331033 - 2981361138811943 x
150388417 x
172824791 x
277174159
Richard Heylen
(9 Aug 2002)
341034 - 291003160575161 x
169865251 x
543834983 x
674139769
Richard Heylen
(9 Aug 2002)
351035 - 7133483249412637 x
577688777 x
800184277 x
867356429
Richard Heylen
(9 Aug 2002)
36(109 - 63)4999999937 x
999999937 x
999999937 x
999999937
Trivial
371037 - 33929611024919407 x
1205624261 x
2704285817 x
2992579621
Jim Fougeron
381038 - 12092131277941309 x
3496358689 x
4329118921 x
5169798247
Jim Fougeron
391039 - 7165012188752617 x
6832015219 x
7332337547 x
9120361379
Jim Fougeron
40(1010 - 33)49999999967 x
9999999967 x
9999999967 x
9999999967
Trivial
411041 - 17521491 1665636553 x
1 2020765051 x
1 2486083933 x
5 7112766749
Jim Fougeron
421042 - 19689211 6387356763 x
2 9844439783 x
3 5069159969 x
5 8304529779
Jim Fougeron
431043 - 8123573 2266433293 x
6 6076452031 x
6 8130819217 x
6 8842829513
Jim Fougeron
44(1011 - 23)49 9999999977 x
9 9999999977 x
9 9999999977 x
9 9999999977
Trivial
451045 - 102101311 1193045207 x
16 2288480071 x
17 0369552917 x
32 5269018463
Phil Carmody
(26 Oct 2003)
461046 - 55062311 4257545709 x
30 1659452561 x
36 9064621643 x
78 6132506111
Phil Carmody
(26 Oct 2003)
471047 - 1239454725 8602566111 x
53 7704923703 x
78 4384962709 x
91 6840728049
Phil Carmody
(27 Oct 2003)
48(1012 - 11)499 9999999989 x
99 9999999989 x
99 9999999989 x
99 9999999989
Trivial
491049 - 1328811112 3363694533 x
113 2900299593 x
161 9376154261 x
485 2216658021
Predrag Minovic
(25 Apr 2006)
501050 - 1827771113 1830822071 x
229 9071133453 x
462 6382993013 x
830 6622844691
Andreas Höglund
(22 Aug 2006)
511051 - 2839931261 0618693979 x
498 1318597973 x
860 7852782717 x
893 3412874271
Andreas Höglund
(16 Sep 2006)
52(1013 - 29)4999 9999999971 x
999 9999999971 x
999 9999999971 x
999 9999999971
Trivial
531053 - 8535811380 1633011217 x
1510 2457237171 x
1535 9336773961 x
3123 5570830697
Richard Heylen
541054 - 55265811503 9932275681 x
2844 1324348441 x
4448 4945266719 x
5255 2241384381
Donovan Johnson
(26 Sep 2007)
551055 - 13729491999 3325988847 x
7348 5361492133 x
8095 2155039507 x
8407 8644846843
Donovan Johnson
(26 Sep 2007)
56(1014 - 27)49999 9999999973 x
9999 9999999973 x
9999 9999999973 x
9999 9999999973
Trivial
571057 - 454110308 6207782083 x
15217 9523379973 x
21127 6757911589 x
30171 1242208009
Richard Heylen
(2 Jun 2003)
581058 - 418775114342 8321659551 x
23611 9351144679 x
52398 1514063681 x
56353 0643461601
Donovan Johnson
(26 Sep 2007)
591059 - 1465574131373 6165207561 x
53613 5329528819 x
67203 5000546687 x
88464 5142291223
Donovan Johnson
(26 Sep 2007)
60(1015 - 11)499999 9999999989 x
99999 9999999989 x
99999 9999999989 x
99999 9999999989
Trivial
611061 - 22336827115611 2937602863 x
182156 8051966979 x
198522 8621423119 x
239190 4358614871
Donovan Johnson
(30 Sep 2007)
621062 - 4626803162259 9674057919 x
217200 8813516923 x
307388 4742899533 x
923080 2373690957
Richard Heylen
(17 Sep 2003)
631063 - 34575809323756 4338664067 x
341522 9830176679 x
944546 8489873021 x
957498 6131281247
Donovan Johnson
(9 Oct 2007)
64(1016 - 63)4999999 9999999937 x
999999 9999999937 x
999999 9999999937 x
999999 9999999937
Trivial
651065 - 110586391199199 9226304927 x
1419138 5939851301 x
1785036 8215222619 x
3291822 5659149497
Donovan Johnson
(3 Oct 2007)
661066 - 11948691668960 8733528161 x
1887779 1141985067 x
3547100 7677518727 x
8948065 2326297719
Donovan Johnson
(30 Sep 2007)
67
68(1017 - 3)49999999 9999999997 x
9999999 9999999997 x
9999999 9999999997 x
9999999 9999999997
Trivial

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.