4-Brilliant numbers

Alpertron
Math problems
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 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.

Maximal and minimal 2-brilliant numbers

Minimal 3-brilliant numbers

Maximal 3-brilliant numbers

Minimal 4-brilliant numbers

Maximal 4-brilliant numbers

Minimal 5-brilliant numbers

Maximal 5-brilliant numbers

Minimal 6-brilliant numbers

Maximal 6-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**
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)
72	10⁷¹ + 49727073	35480737 2690810359 × 51446094 7863964273 × 58474935 7034783263 × 93688265 3165860553	Eric Jeancolas (14 Nov 2022)
73	(10¹⁸ + 3)⁴	100000000 0000000003 × 100000000 0000000003 × 100000000 0000000003 × 100000000 0000000003	Trivial
74	10⁷³ + 59203473	103308079 9530308497 × 189374249 3777406457 × 195283966 2287012731 × 261744882 0210223627	Eric Jeancolas (17 Nov 2022)
75	10⁷⁴ + 5773243	137595665 3530253917 × 277293315 6399582559 × 296346535 4433379559 × 884414781 2502586159	Donovan Johnson (26 Sep 2007)
76	10⁷⁵ + 59947983	331491739 8958401133 × 464252389 1272309817 × 679310629 0817007599 × 956543514 5508017197	Eric Jeancolas (30 Dec 2024)
77	(10¹⁹ + 51)⁴	1000000000 0000000051 × 1000000000 0000000051 × 1000000000 0000000051 × 1000000000 0000000051	Trivial
78	10⁷⁷ + 81290677	1340154745 0483415969 × 1587883849 5696816989 × 2098656276 7140434843 × 2239159371 9739394779	Eric Jeancolas (3 Mar 2025)
79	10⁷⁸ + 3551977	1139707345 3742002073 × 3588019863 3662935129 × 4939950664 3173123821 × 4950273594 7329553661	Eric Jeancolas (30 Nov 2024)
80	10⁷⁹ + 162942837	2405086113 2118000673 × 6857163865 1164536347 × 7771816923 8492009057 × 7801934137 2305521711	Eric Jeancolas (29 Nov 2025)
81	(10²⁰ + 39)⁴	1 0000000000 000000039 × 1 0000000000 000000039 × 1 0000000000 000000039 × 1 0000000000 000000039	Trivial
82	10⁸¹ + 36986263	1 3495118169 6477412879 × 1 8417458011 5291954613 × 1 8788134778 0288680621 × 2 1414599316 2584616489	Eric Jeancolas (24 Jul 2025)

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**
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 − 8497353	1195229 7755351279 × 1535049 4322155981 × 1977694 7922547331 × 2755922 2491901663	Eric Jeancolas (20 Sep 2024)
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)
71	10⁷¹ − 130515921	35357569 5372020413 × 41223644 8305055397 × 79928142 4818569639 × 85836398 6232321001	Eric Jeancolas (11 Dec 2024)
72	(10¹⁸ − 11)⁴	99999999 9999999989 × 99999999 9999999989 × 99999999 9999999989 × 99999999 9999999989	Trivial
73	10⁷³ − 16262381	137788056 3929101801 × 170507812 9301062601 × 192091554 2407174283 × 221582711 0111357993	Eric Jeancolas (3 Nov 2022)
74	10⁷⁴ − 6161223	179231100 1058017763 × 213228118 2850024163 × 457237580 9735857993 × 572269021 2161215081	Eric Jeancolas (29 Oct 2022)
75	10⁷⁵ − 60215153	264680180 0234228867 × 508016058 0032915219 × 801673834 7206051711 × 927691183 5372089849	Eric Jeancolas (23 Nov 2022)
76	(10¹⁹ − 39)⁴	999999999 9999999961 × 999999999 9999999961 × 999999999 9999999961 × 999999999 9999999961	Trivial
77	10⁷⁷ − 37981169	1178767491 2293349149 × 1409858606 4958079329 × 2004595514 2799263129 × 3001715555 8413339059	Alfred Reich (18 Apr 2024)
78	10⁷⁸ − 19542569	1456004956 4152162627 × 1514461440 6108482891 × 4878870279 7691376017 × 9295219614 4518059399	Alfred Reich (6 May 2024)
79	10⁷⁹ − 1439099	2087009745 5121334319 × 5096003848 8655265483 × 9513638525 4138554017 × 9883235168 9036406689	Alfred Reich (4 May 2024)
80	(10²⁰ − 11)⁴	9999999999 9999999989 × 9999999999 9999999989 × 9999999999 9999999989 × 9999999999 9999999989	Trivial
81	10⁸¹ − 114497433	1 2823190258 2529223349 × 1 8784647198 8333836201 × 1 9571967280 6565514883 × 2 1211256225 6063562001	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.

4-Brilliant numbers

Table of contents

Minimal 4-brilliant numbers

Maximal 4-brilliant numbers