4-Brilliant numbers

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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.

Table of contents

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.