Brilliant numbers

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Math problems
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)

Maximal and minimal 2-brilliant numbers

The next table contains the maximal and minimal numbers with given number of figures whose two prime factors have the same length when expressed in decimal notation.

**Maximal and minimal 2-brilliant numbers**
Digits	Number	Prime factors	Discoverer
1	9	3 × 3	Jim Fougeron
2	10	2 × 5	Dario Alpern
3	989	23 × 43	Dario Alpern
4	1003	17 × 59	Dario Alpern
5	99973	257 × 389	Dario Alpern
6	100013	103 × 971	Dario Alpern
7	10⁷ − 189	1583 × 6317	Dario Alpern
8	10⁷ + 43	2089 × 4787	Dario Alpern
9	10⁹ − 137	25303 × 39521	Dario Alpern
10	10⁹ + 81	26881 × 37201	Dario Alpern
11	10¹¹ − 357	231571 × 431833	Dario Alpern
12	10¹¹ + 147	281683 × 355009	Dario Alpern
13	10¹³ − 77	1380163 × 7245521	Dario Alpern
14	10¹³ + 73	1857929 × 5382337	Dario Alpern
15	10¹⁵ − 261	18874013 × 52982903	Richard Heylen
16	10¹⁵ + 3	14902357 × 67103479	Dario Alpern
17	10¹⁷ − 297	150425719 × 664779937	Richard Heylen
18	10¹⁷ + 831	158267567 × 631841393	Dario Alpern
19	10¹⁹ − 533	1809444697 × 5526557411	Richard Heylen
20	10¹⁹ + 49	1172005397 × 8532383917	Dario Alpern
21	10²¹ − 353	2 1942092327 × 4 5574505161	Richard Heylen
22	10²¹ + 987	1 0119972739 × 9 8814495433	Dario Alpern
23	10²³ − 587	19 8168943561 × 50 4619937933	Richard Heylen
24	10²³ + 691	25 3850475563 × 39 3932687257	Dario Alpern
25	10²⁵ − 113	270 1224339427 × 370 2024986981	Richard Heylen
26	10²⁵ + 183	163 4713216901 × 611 7280937483	Dario Alpern
27	10²⁷ − 281	1681 6039284931 × 5946 7035195149	Richard Heylen
28	10²⁷ + 4153	1059 4044136579 × 9439 2659413907	Dario Alpern
29	10²⁹ − 93	27335 0346504617 × 36583 0888011371	Richard Heylen
30	10²⁹ + 279	26292 2036885471 × 38034 0884258249	Dario Alpern
31	10³¹ − 389	185693 5855186693 × 538521 5634707327	Richard Heylen
32	10³¹ + 667	117816 0270652879 × 848780 9552819573	Dario Alpern
33	10³³ − 1083	2758682 6038690693 × 3624918 6426792769	Richard Heylen
34	10³³ + 709	2856715 9477900867 × 3500523 0421091927	Dario Alpern
35	10³⁵ − 783	29079576 8932439557 × 34388395 8033904381	swishzzz (11 Oct 2020)
36	10³⁵ + 277	31231547 3725119131 × 32018906 6546263567	Jim Fougeron
37	10³⁷ − 707	112407394 0762336703 × 889621192 8210070531	Richard Heylen
38	10³⁷ + 1687	291898446 0772347259 × 342584900 1386617493	Jim Fougeron
39	10³⁹ − 1071	2216872853 9128208483 × 4510858609 8430581563	Richard Heylen
40	10³⁹ + 997	1828883498 3913311471 × 5467816844 9744921707	Jim Fougeron
41	10⁴¹ − 3699	1 2706241645 7894753317 × 7 8701478208 6546036553	Richard Heylen
42	10⁴¹ + 1207	2 2872845388 3196320979 × 4 3719965007 5309509133	Jim Fougeron
43	10⁴³ − 27	25 5149936898 7998869591 × 39 1926414779 6477703203	Richard Heylen
44	10⁴³ + 91	20 9416220529 8508378983 × 47 7517929351 3497904877	Jim Fougeron
45	10⁴⁵ − 17	153 8012743463 0196194863 × 650 1896712171 4506226241	Richard Heylen
46	10⁴⁵ + 1411	204 5492562244 7906868371 × 488 8798025755 5528444241	Jim Fougeron
47	10⁴⁷ − 1677	2791 7031852805 8249521813 × 3582 0426944832 7527799271	Richard Heylen
48	10⁴⁷ + 393	2055 2256009243 0172482139 × 4865 6458908952 2586263787	Jim Fougeron
49	10⁴⁹ − 471	19436 4326829329 0666134453 × 51449 7704549507 1915161893	Richard Heylen
50	10⁴⁹ + 951	26471 4152186265 7404330791 × 37776 5975767080 0343659761	Jim Fougeron
51	10⁵¹ − 579	291216 6130819063 0398411623 × 343387 0030343167 2610407227	Richard Heylen
52	10⁵¹ + 9793	139533 7722542092 6914498491 × 716672 3753287142 5027689523	Jim Fougeron
53	10⁵³ − 167	1858626 3518991829 5875711699 × 5380317 5607522149 7635484067	Richard Heylen
54	10⁵³ + 2217	1408301 9631082166 1891328463 × 7100749 8831637862 7095268359	Jim Fougeron
55	10⁵⁵ − 267	10492309 5218034300 4616980607 × 95307901 2701599053 8677972619	Richard Heylen
56	10⁵⁵ + 6229	14602019 9335556486 0764167713 × 68483675 8578849654 5968810933	Jim Fougeron
57	10⁵⁷ − 8349	244705702 9837201547 5229439271 × 408654145 6970164126 1618151781	Jim Fougeron
58	10⁵⁷ + 2317	218276491 2198936605 3485370671 × 458134540 4680301502 5118660227	Jim Fougeron
59	10⁵⁹ − 5973	1009891100 3114765934 0436549999 × 9902057753 4703898106 0013155973	Jim Fougeron
60	10⁵⁹ + 213	2139168817 8982927891 0570173437 × 4674712868 0684854707 6331371449	Peter Wallrodt
61	10⁶¹ − 69	1 0552178828 8211523215 4201320731 × 9 4767158159 6685321140 0570143201	Bouk de Water
62	10⁶¹ + 399	1 9605820850 2188312577 4065415291 × 5 1005260511 1322566578 9755534589	Bouk de Water
63	10⁶³ − 9	14 4992163448 9889675295 9323769763 × 68 9692446965 8108371268 4424134557	Bouk de Water
64	10⁶³ + 19	22 3505686289 8037778009 5228766733 × 44 7415909903 6844148191 4582608543	Jim Fougeron
65	10⁶⁵ − 687	158 9971766331 0591078685 6570509671 × 628 9419857483 0732390318 5090439303	Bouk de Water
66	10⁶⁵ + 2317	286 5952168341 3346502850 9515988163 × 348 9241764208 3274582598 6935358959	Jim Fougeron
67	10⁶⁷ − 809	1250 5110651609 5154757791 7550796961 × 7996 7305197038 8910336438 6584269431	Jim Fougeron
68	10⁶⁷ + 609	1678 0927835184 8046981335 7050416541 × 5959 1460604656 5611794502 9500096149	Jim Fougeron
69	10⁶⁹ − 1241	10679 9049628454 5686213838 6056396401 × 93633 7920120938 1211489257 1488571159	Phil Carmody
70	10⁶⁹ + 2607	24426 5973536430 8480998741 7412256391 × 40938 9807971291 5155256416 4844215577	Jim Fougeron
71	10⁷¹ − 7619	231008 2702166394 9461270808 8690827091 × 432884 9348389996 0157713857 5168198191	Jim Fougeron
72	10⁷¹ + 11901	162435 9550602349 2398975054 0378209139 × 615627 2480616606 0652634801 6196129359	Jim Fougeron
73	10⁷³ − 8297	2279546 3843252090 7465709063 0001002097 × 4386837 6922543729 0923364184 1447115399	Richard Heylen
74	10⁷³ + 10563	1093872 1328861786 2529696746 2523488383 × 9141836 3256179014 2787369950 1869702461	Jim Fougeron
75	10⁷⁵ − 4209	11603482 2931805642 1367482412 2706652977 × 86181025 2072092157 5415636921 1739413183	Richard Heylen
76	10⁷⁵ + 5473	11171602 0162596298 9233595908 9815716691 × 89512676 7445310892 2620197682 3759569403	Jim Fougeron (2 May 2002)
77	10⁷⁷ − 4851	266574540 5469304796 5242036511 4117132441 × 375129597 1281810690 2056255934 4084771989	Richard Heylen (10 May 2002)
78	10⁷⁷ + 3	185554311 4966205323 7177035161 1143673123 × 538925768 9214151307 3970633306 9119686561	Jim Fougeron
79	10⁷⁹ − 5667	1729776145 3696959942 8207517694 6198769503 × 5781094869 8582916253 8759307497 1223276611	Richard Heylen
80	10⁷⁹ + 5923	1333322076 5188990013 5038176080 7974795003 × 7500063320 1157802123 7780289418 0923803641	Richard Heylen (16 May 2002)
81	10⁸¹ − 4989	2 8090949127 0712637360 1146061968 3106498869 × 3 5598654765 1499403487 7251052333 5003778519	Michael Bell
82	10⁸¹ + 17527	1 6185521526 1918373528 4137894011 0747187933 × 6 1783613112 6064522717 8709752383 9696139619	Richard Heylen
83	10⁸³ − 14333	19 9932327974 2988610690 4886345531 6009796077 × 50 0169237327 4167000023 1182095228 9282634671	Richard Heylen (27 Jun 2002)
84	10⁸³ + 8569	13 2087033835 8319180791 0364106785 8862243541 × 75 7076581220 5898287201 5989811643 3821618709	Richard Heylen (27 Jun 2002)
85	10⁸⁵ − 7007	169 3200872034 1695875408 3626191779 6700786223 × 590 5973806868 0815745483 1212278304 4390802991	Richard Heylen (8 Jul 2002)
86	10⁸⁵ + 16701	305 1275553674 2722241262 5346271954 4495568863 × 327 7317903313 6623331914 1570013682 9929609827	Richard Heylen (27 Sep 2002)
87	10⁸⁷ − 6113	1672 0583183841 0336460297 3032415415 4355596829 × 5980 6526423457 0267994286 2190355537 5545514603	Richard Heylen (11 Oct 2002)
88	10⁸⁷ + 11989	2554 8073574973 3420228780 0723585352 7655387529 × 3914 1894478478 0487593972 6201920459 1690357741	Reto Keiser (25 Jul 2002)
89	10⁸⁹ − 14547	17182 5942024805 9924461238 7996462002 8962720089 × 58198 4296559615 5552330942 9349240261 7340927477	Richard Heylen (24 Nov 2002)
90	10⁸⁹ + 9757	12544 9117210655 5918816731 8368413723 3747818249 × 79713 5940240048 5345670349 5028670126 7149637493	Richard Heylen (24 Nov 2002)
91	10⁹¹ − 35579	297819 0131233156 6877668999 3983804239 6357404141 × 335774 3985223460 3802184405 8109031708 9334569081	David Cleaver (5 Sep 2002)
92	10⁹¹ + 6489	303042 6176188825 9047082534 8078377909 3519047793 × 329986 5899579960 5435086827 9346278922 2873063273	David Cleaver (11 Nov 2002)
93	10⁹³ − 13643	1308651 9402050960 4707836850 1221022392 5173521497 × 7641451 2467178771 4260836429 3152368823 0774568381	David Cleaver (9 Dec 2002)
94	10⁹³ + 3489	2941166 7593005444 7010239755 9827007826 4073129867 × 3400011 2262856379 6700171396 6956849633 0607787267	Jim Fougeron (9 Dec 2002)
95	10⁹⁵ − 3681	11109749 1776794870 9376300022 4384096761 6045495643 × 90011033 0131568068 2277608653 0971942745 8876830733	Richard Heylen (30 Jan 2003)
96	10⁹⁵ + 2899	13630093 2834042408 3940245923 3530210953 6892634161 × 73367069 4108588540 8146519774 5857415052 8292043459	Jim Fougeron (12 Dec 2002)
97	10⁹⁷ − 19163	150730263 1098002948 3909591311 6970605940 8828799543 × 663436777 3056592241 5154945278 4246524717 6344374659	David Cleaver (20 Mar 2003)
98	10⁹⁷ + 12577	124825954 2695921544 0349409041 9477333875 1647883781 × 801115445 7832187771 2536036138 9540856834 1998334317	Jim Fougeron (3 Jan 2003)
99	10⁹⁹ − 2783	3108480161 2134411476 3336994966 2780217266 0847517087 × 3217006215 6987845422 4355815533 0840713863 2546972991	David Cleaver (13 Dec 2002)
100	10⁹⁹ + 4011	2315648505 7650669192 7684971633 4055819213 0875875893 × 4318444692 7518910400 6475667703 3452093865 5251359327	David Cleaver (20 Dec 2002)
101	10¹⁰¹ − 56993	1 1782454340 0831857004 2642818908 5916894417 2987388481 × 8 4871960555 6256173460 8988925131 0824028091 2733932447	David Cleaver (4 Mar 2003)
102	10¹⁰¹ + 3231	2 2701919653 5950614963 1764979892 5506904523 4236491673 × 4 4049138366 2192037893 8689413501 3226586083 9934720247	Richard Heylen (13 Jul 2003)
103	10¹⁰³ − 2417	23 8868404401 3249161833 4280106561 3978194751 1337567461 × 41 8640549178 6561972002 2578515523 6366529229 1130223203	David Cleaver (20 Jul 2003)
104	10¹⁰³ + 17389	21 7885032974 4203343440 1041331778 9198161891 9402435883 × 45 8957637589 2693047888 0734959940 6651271487 1682387783	David Cleaver (17 Aug 2003)
105	10¹⁰⁵ − 30179	200 7284510838 9786097528 9736163621 0062604835 7754871711 × 498 1854812310 7522887110 5253775420 8478809696 2615971011	Richard Heylen (14 Oct 2003)
106	10¹⁰⁵ + 4293	128 4226997750 6686435684 4426458148 2168133598 8332440829 × 778 6785371678 8858218366 8785177552 7053995002 6013843817	Richard Heylen (1 Aug 2003)
107	10¹⁰⁷ − 569	2981 0893150893 5388275225 5462174328 6502046414 6509030899 × 3354 4784952879 7640933380 6165775647 0675149353 0526472669	Richard Heylen (22 Oct 2003)
108	10¹⁰⁷ + 6969	2000 1790349395 8497496269 8258346499 2284082246 7385473869 × 4999 5524527143 3818753431 7433455459 0767522642 1199679901	Richard Heylen (30 Oct 2003)
109	10¹⁰⁹ − 10617	29392 9347145171 5027656817 5741564598 1896353292 9889785991 × 34021 7814149092 3281380298 5431644803 2470409630 1595136513	Richard Heylen (30 Nov 2003)
110	10¹⁰⁹ + 19041	27622 9153544139 4586840378 2275819821 0837314064 7210683969 × 36201 8268951545 3682000526 8087291994 7197343843 5837408289	David Cleaver (16 Nov 2003)
111	10¹¹¹ − 11957	141182 0397984271 8112107179 0859135842 7040257882 1772762747 × 708305 3916969546 1569685870 3508927026 7835105482 4192755569	Richard Heylen (5 Apr 2004)
112	10¹¹¹ + 57339	162769 0923050353 6141393381 7404912003 9148307877 6043167501 × 614367 2523073131 3013378040 0848273214 4910301727 0012524839	Sander Hoogendoorn (15 Mar 2006)
113	10¹¹³ − 91227	1354837 3134821631 0599104139 9619149091 5690067442 5643544869 × 7380959 9872166890 9879514364 9871648779 4623234653 1808934017	Sander Hoogendoorn (20 Apr 2006)
114	10¹¹³ + 29359	1299567 8224153860 6843279942 5018706158 2368298838 8592729389 × 7694865 8065524647 5123803852 8936909680 8671673449 0830614731	David Cleaver (27 Mar 2007)
115	10¹¹⁵ − 1769	11504120 0967794696 9526346072 3317602333 5486779937 1253396127 × 86925379 0457164835 6378342160 3635526344 0656363176 0612220553	David Cleaver (1 Apr 2007)
116	10¹¹⁵ + 12741	16593409 1113277430 7356923305 1068375914 2603433470 9864681011 × 60264891 5175203383 9345332361 8479362122 0593165136 5703227431	Sander Hoogendoorn (28 Apr 2006)
117	10¹¹⁷ − 2733	157262582 8389974101 2641225377 7830114415 2030142372 3971096511 × 635879165 8812967134 9443296973 9980973373 1062341237 4598098797	David Cleaver (18 Dec 2007)
118	10¹¹⁷ + 7341	150501763 2843814481 3324771965 6235019180 5691855629 0873576027 × 664444042 4996512565 1345348476 3734158530 5429480908 1436607383	David Cleaver (2 Oct 2008)
119	10¹¹⁹ − 19107	1060245251 4498526274 2887581992 2502026799 9483770339 0958334023 × 9431780039 8778574370 9001406775 6729772644 6849617817 0211877691	David Cleaver (16 Oct 2008)
120	10¹¹⁹ + 10443	1733519903 6894877000 6118014625 4204082180 9720757108 9955436713 × 5768609854 8489607151 8016590923 2695970673 1224747267 5097328211	David Cleaver (12 Apr 2010)
121	10¹²¹ − 9633	1 3679005690 1172640709 6267890378 4274826594 0319877496 5236311639 × 7 3104728710 1046188617 2597258203 0918600850 5948605983 0493386553	David Cleaver (13 Jan 2009)
122	10¹²¹ + 7441	1 6050549491 5706436498 3042540776 2178545228 9715708181 8323775159 × 6 2303162924 4329337156 1630045207 4654938693 3502634075 8295841399	David Cleaver (27 Mar 2010)
123	10¹²³ − 26453	28 0732002176 0375200288 1036939514 1204456905 1168037124 4497994669 × 35 6211615437 0366033759 4063098262 2266482771 6307268511 0342932663	David Cleaver (8 Apr 2010)
124	10¹²³ + 7051	23 4808021163 5120204597 3315154645 5783029514 2835223140 1178333959 × 42 5879829421 8557747573 1058591303 5641444527 8649483795 6270583389	David Cleaver (23 Apr 2010)
125	10¹²⁵ − 16557	166 5703358155 1519697547 2443830050 4891338492 9904334727 2954825491 × 600 3469916201 3395047367 9222482756 5444694530 9809096415 7383508673	David Cleaver (27 Apr 2010)
126	10¹²⁵ + 4521	132 2106277541 6234517159 3168355643 6144397989 2156046465 1676992659 × 756 3688464284 7278050326 7458846713 2288251246 4805056433 7149416019	David Cleaver (23 Oct 2012)
127	10¹²⁷ − 2907	3004 6146242548 0114175254 8298266004 2163681505 6844271964 0543520887 × 3328 2138478841 2961926743 3103503463 1111424047 7541722530 5832697539	David Cleaver (24 Oct 2012)
128	10¹²⁷ + 13009	1646 5124577863 8822072517 3031177402 2736202511 3306978968 2876014669 × 6073 4432665296 9607988725 3145345114 1404853786 2317634215 3392307861	David Cleaver (27 Oct 2012)
129	10¹²⁹ − 55949	10064 3878552488 4862571188 0039470176 5686284645 4651272528 7405454369 × 99360 2407203010 5010756776 9406969213 2116735641 9436356113 7033148179	David Cleaver (11 Nov 2012)
130	10¹²⁹ + 123	11664 4308485686 3136962963 7442720892 0783290888 4771675415 5425408347 × 85730 7152815529 1490396446 6215480820 8096809638 2724778190 7539975009	David Cleaver (29 Apr 2010)
131	10¹³¹ − 92459	197427 0546102759 0950305557 3039977052 8045679849 8936939142 7647811271 × 506516 1925117181 2520125248 1400143562 4573028824 5245558499 6603306371	David Cleaver (28 Dec 2013)
132	10¹³¹ + 91041	109528 5074437630 8645546283 5362783529 9757711413 9912969995 9296437089 × 913004 3158064994 6109965404 6315029952 2026636572 6210861515 1663407169	David Cleaver (29 Sep 2014)
133	10¹³³ − 21099	3071414 3518267268 7378766911 9136020019 8591833148 9211667886 5902179259 × 3255829 0267975369 7845781898 1553165790 6568164764 8501887629 9731904239	David Cleaver (25 Dec 2014)
134	10¹³³ + 2413	1494152 8028372203 6615875084 9004183379 1363514410 0174401041 0728805027 × 6692755 9089078283 2218859274 9508639420 2028395439 6040525253 6984044719	David Cleaver (26 Dec 2014)
135	10¹³⁵ − 431	20806007 8237600471 7798336653 1024716689 8008684206 2349960596 6016419309 × 48063040 6597280948 0574777434 5208478847 9150683268 9533275897 5323553141	David Cleaver (30 Apr 2010)
136	10¹³⁵ + 18463	21274932 0382385573 4511834899 4979437872 2737173093 9014203645 5070066967 × 47003675 4149271669 6885226327 0099126966 9770466918 1504850773 9406495289	David Cleaver (1 Jan 2015)
137	10¹³⁷ − 23393	115605278 7860292516 9798614498 3947697677 1775723061 5640765963 5859576889 × 865012402 9810727860 9987442135 1008178560 7659325917 8752159424 1686893463	David Cleaver (9 Jan 2015)
138	10¹³⁷ + 17841	163776634 7045626420 0924850769 4981115586 2764208395 3675453495 9460390339 × 610587707 9498575560 4475869051 0189671325 9966606123 4102523721 6625482619	David Cleaver (14 Jan 2015)
139	10¹³⁹ − 32771	1458143858 1014743140 0544917914 0802992633 4618724206 1788926822 0382130409 × 6858033893 1853771284 4989669450 1312471774 6263440159 1184123680 9507604981	David Cleaver (27 Jan 2015)
140	10¹³⁹ + 11389	1711572707 2776814018 8265343657 6961984319 6281408606 9774557920 1843022487 × 5842579726 5167679908 1469224446 6661242985 4045892595 9249543069 6082753547	David Cleaver (14 May 2010)
141	10¹⁴¹ − 45669	1 2113021658 3714674710 4649985070 6246065013 4548495912 8673352637 6480157581 × 8 2555784031 7149057049 4923559496 5960829095 3216319219 4268567086 5728631751	David Cleaver (6 May 2015)
142	10¹⁴¹ + 33453	3 1124911728 2728238224 2148600314 9122659266 4573224320 3951640140 2032184271 × 3 2128605174 2159193418 4519913284 1620333856 1063999436 1079097157 2320283843	David Cleaver (17 Apr 2016)
143	10¹⁴³ − 23841	13 5338143715 8648047268 9238134266 1682230201 4793595653 9350708690 7358098971 × 73 8889992535 6199085636 4480273022 6772032485 7281496718 7321971695 9251686029	David Cleaver (17 Apr 2016)
144	10¹⁴³ + 82821	13 0346521715 3628801868 7180980785 8006804650 2667829251 3682235182 3650537987 × 76 7185795861 6613897500 5269616708 2645623225 4543107978 5932201621 0304397783	David Cleaver (17 Apr 2016)
145	10¹⁴⁵ − 40067	214 3224364496 0829700335 7241579771 3808175407 4948412018 1981471914 4014601703 × 466 5867076567 6975118944 8254269956 6223786911 3542281868 7666926472 8008220411	David Cleaver (11 Mar 2011)
146	10¹⁴⁵ + 26019	171 2231579162 6950231464 2400513436 2656947458 2230088593 8520006217 5608237361 × 584 0331484185 1816669465 2639928342 6386742617 2203932732 7824314625 2757154579	Ben Whitmore (8 Sep 2019)
147	10¹⁴⁷ − 28239	1074 9227813857 8128425606 3886603957 1941385856 7534807145 3665042338 7382761653 × 9302 9938272478 3752992029 5079341383 9971743799 1436851707 8403961821 6410820237	Alfred Reich (26 Sep 2020)
148	10¹⁴⁷ + 19	1121 0754325987 9038482223 5833349894 6062147584 2503429633 0596159095 2868977237 × 8920 0063699717 0940867415 7068624024 3731188867 2010102299 2106066640 6453209287	Andreas Höglund (29 Jul 2006)
149	10¹⁴⁹ − 81	20049 5535838339 3829556681 4309092933 2219817874 2132003916 5825052578 7307984071 × 49876 4222264931 2847094265 1352367968 9583086107 5206340391 9176400310 7273225689	Serge Batalov (19 Nov 2014)
150	10¹⁴⁹ + 21553	19617 5124573517 0920349224 9342247090 8957491913 2952604732 0351896976 4669282857 × 50974 8624946200 8799446688 3520101722 0741864841 3016354748 0040592011 5982269129	Paul Leyland (6 Jul 2006)
151	10¹⁵¹ − 13731	295723 2918876522 6188100992 9602747628 7595656664 8216412474 4313834384 9506540789 × 338153 9525063545 9482199099 8152412930 1893159430 7868919366 6457905073 9840837321	Alfred Reich (29 Sep 2020)
152	10¹⁵¹ + 14541	189206 6823280306 9424026723 9983662103 6921105169 1017024404 1717170855 6479073929 × 528522 5594021482 7041716434 2787336881 1118569345 4236208742 0343844568 9349531429	swishzzz (11 Oct 2020)
153	10¹⁵³ − 24783	1961682 4731935528 3234134293 2321794612 6629367330 5583782958 2802414444 2130486423 × 5097664 9568165522 9290563513 8882236597 3469595971 1655609204 7397403736 1186978679	Alfred Reich (6 Oct 2020)
154	10¹⁵³ + 6459	1656183 3025918989 6336397733 2001682090 8206964193 4150579667 2224175450 7702593741 × 6037978 9992751215 9491364699 7651810129 4076069067 1891373656 5370593550 4604572199	Eric Jeancolas (15 Mar 2021)
155	10¹⁵⁵ − 70301	13748346 0682971758 2310178558 1604369932 9165454253 3072883571 3337978179 4162674443 × 72736021 8481797830 1985507698 8283737498 3036932333 9204705211 6566195061 2603395593	David Cleaver (10 Sep 2012)
156	10¹⁵⁵ + 7213	21174722 4607852036 3337301814 8076886231 7143398243 2276510817 7550584803 1153848949 × 47226120 7603529495 5119132841 4854538873 5587409943 7205441484 2256823065 0370558937	swishzzz (19 Oct 2020)
157	10¹⁵⁷ − 2049	144681472 7739974723 8275556170 2787620697 7909532017 9427341752 9475932384 7644834853 × 691173500 5366371832 7873851967 0806005864 7586415356 7211937810 4789158248 8320067667	Alfred Reich (27 Nov 2020)
158	10¹⁵⁷ + 35923	293515961 3287504553 9293785709 0898199998 3260302586 4214269938 9786675642 9269144899 × 340696974 5266960609 1149680695 1378800110 3961332546 4154535826 9798584353 8313491377	Eric Jeancolas (22 Mar 2021)
159	10¹⁵⁹ − 14423	2547072684 4680018375 1450575263 3857339593 7158018283 5125296043 2398227515 4411906263 × 3926075632 2266731944 9647556524 9617102704 5569589355 6783742423 3435239098 9343991679	Eric Jeancolas (19 Feb 2021)
160	10¹⁵⁹ + 19551	1669042030 6222239980 9533336540 3271748157 0580844804 3833871582 7253914560 0831962237 × 5991460859 8993575190 6841900164 5932717847 5252288991 6792804522 4715169782 3138541323	Eric Jeancolas (5 Feb 2021)
161	10¹⁶¹ − 687	2 6896817575 3118233694 3149550685 3260830491 6004144769 2517229652 1971743723 9219671929 × 3 7179119693 2489393821 8095105140 9845272240 8958423031 5780414610 9392239191 2317151897	Eric Jeancolas (12 Feb 2021)
162	10¹⁶¹ + 4047	1 8932724353 2053672714 1435616197 6577272977 0482897654 5749205957 4040469712 2172363041 × 5 2818600289 3290414018 3437718744 8715615330 3956513444 6098351258 3770032398 9924848367	Eric Jeancolas (29 Jan 2021)
163	10¹⁶³ − 819	15 4049565409 0008958982 0798656941 6142371880 8031252049 8620643349 6256604447 1439562841 × 64 9141720942 2204766999 9784534709 0616054256 4870739731 2560352818 5761699900 1845774741	Eric Jeancolas (1 Feb 2021)
164	10¹⁶³ + 1759	28 0083815480 3505867546 1948724562 6166645818 3551750082 8781645017 7219206254 8119466379 × 35 7035981634 6316074325 5785803361 8875322598 7108799268 6730832102 5975235826 0151108221	Eric Jeancolas (29 Jan 2021)
165	10¹⁶⁵ − 27557	168 2965066534 0802699068 4368622418 5580955103 0170255430 7712544205 1067729888 3957782921 × 594 1893981551 3387837458 5994150609 3792970941 9123971038 4754476751 0563208460 7269190083	Erik Branger (21 Nov 2019)
166	10¹⁶⁵ + 21577	171 0894728567 6514420334 2348472529 2468915931 6004710883 6576330874 9481449642 5420952639 × 584 4894973971 8509083458 7550765132 0000921411 9314533674 7653498401 0612632218 3596397943	Erik Branger (14 Oct 2019)
167	10¹⁶⁷ − 38903	2032 9567951828 0624355545 6161681508 6049996967 1339902409 7109146581 9581104012 2874591267 × 4918 9436901440 8217255986 8212888481 4429349123 2238922901 4681138054 0323859898 7818347491	Erik Branger (1 May 2020)
168	10¹⁶⁷ + 6453	2032 1491344864 1292965085 6141338757 8482611027 1627178496 3341645623 8628001836 0230767193 × 4920 8986832195 8070178727 5161574093 9774394038 6446977363 0884742431 4512350765 2778037821	Erik Branger (15 Dec 2019)
169	10¹⁶⁹ − 14319	20939 6376022990 9907466815 0252921445 7776796197 2509185032 1325968652 6778149196 8551925027 × 47756 3183753573 4107517020 0486845194 0980286251 8997809812 0353070711 4410818249 6827007803	Erik Branger (1 Nov 2020)
170	10¹⁶⁹ + 25831	15786 4055332270 6420836164 8929652825 2651079583 3878698113 1064320745 4453202028 7270837641 × 63345 6424196889 0714235608 0693374664 2207264980 1566867783 8070883917 7126196228 6575780591	Erik Branger (30 Aug 2020)
171	10¹⁷¹ − 16569	100260 7307437205 3022855343 7496173168 3656654844 8825765741 8686914675 2951415541 8582501607 × 997399 4729363484 1017115301 3012960532 6405313615 0611286806 8310260187 7611566448 2913987233	Erik Branger (10 Dec 2020)
172	10¹⁷¹ + 7467	159823 3917065448 8061693029 1400065214 0081240734 8641102533 4770714446 4074697295 5602480993 × 625690 6384743237 1483112919 2492406945 7572438680 7642240564 8158440979 7982147224 3476190219	Erik Branger (13 Nov 2020)
174	10¹⁷³ + 46323	2196785 4151894359 2018357235 8106966210 3853458688 8886266590 6327872506 8869329739 2179181097 × 4552105 9685011007 8552386152 1399905274 4182864898 1499480797 6987988036 9451182155 2768572859	Erik Branger (19 Jul 2021)
180	10¹⁷⁹ + 1039	1408377255 6390310892 8160798541 4167793439 8706910170 6278981482 4520862904 3783377250 3658895889 × 7100370273 6328574475 1511636041 0309625329 5689143616 1203659956 8602760227 7423489536 8099896351	fivemack (11 Sep 2019)
181	10¹⁸¹ − 28457	1 9767809959 0457526282 4882988600 3773841327 2058367732 8095507348 8102732219 8684412758 6224818989 × 5 0587293284 9802038293 7746413638 9678419339 8327136055 8098096714 1424015651 6919173312 2358525587	Erik Branger (19 Sep 2022)
182	10¹⁸¹ + 38439	1 0543000600 0073005648 8062242818 1216612707 8688770654 7586912142 4546411916 0566738098 3978269333 × 9 4849657885 7547958328 5807708150 9813977229 3150003505 4082588713 9477534678 1747511786 3857748683	Erik Branger (19 Sep 2022)
199	10¹⁹⁹ − 5399	1665689442 2236107849 2461170014 6708336391 5297020262 6236209523 3433137957 4662666048 9754819717 3600787241 × 6003520071 9352030115 1305968163 7525147850 0171431803 2268401059 2432041391 0731400891 4754541467 5783616961	Alfred Reich (21 Nov 2020)

Names in italics denote usernames of Mersenneforum forum members.

Density of brilliant numbers (by Richard Heylen)

Instead of deriving density results using other people's work I decided to think about the problem myself for a change and I'm quite pleased with the results.

With brilliant numbers we're trying to represent a number in the region of bⁿ (where n is odd) as the product of two primes where the larger of the two primes ranges from b^n/2 to b^(n+1)/2. First we work out the average number of factors that a number bⁿ ± x has in this range. If we say that the factor in the range b^(n-1)/2 to b^n/2 is p and the larger factor is q. It's easy to see that they have the same number of digits base b. If we plot p against q then the number of ways is the number of lattice points on the curve p = (bⁿ+x)/q in the relevant region. We can estimate this by looking at the area between the curves p = bⁿ/q and p = (bⁿ+1)/q between the specified limits. I make this to be roughly (ln b)/2 which, for base 10 is about 1.15.

I initially found it surprising that a number would be representable as the product of two numbers in the relevant ranges in on average more than one way. After all, some of these numbers are prime or are the product of a tiny prime and a large prime so I looked a bit closer.

1000 = 10 × 100 = 20 × 50 = 25 × 40
1001 = 11 × 91 = 13 × 77
1002 = ...
1003 = 17 × 59
1004 = ...
1005 = 15 × 67
1006 =...
1007 = 19 × 53
1008 = 12 × 84 = 14 × 72 = 16 × 63 = 18 × 56 = 21 × 48 = 24 × 42 = 28 × 36
1009 = ...
1010 = 101 × 10
1011 = ...
1012 = 11 × 92 = 22 × 46 = 23 × 44
1013 = ...
1014 = 13 × 78 = 26 × 39
1015 = 39 × 35
1016 = ...
1017 = ...
1018 = ...

So that's 19 numbers representable in a total of 22 ways which works out to an average of 1.158 which is reasonable agreement. We then have to make sure that both p and q are prime. I can make a handwaving argument for the probabilities being independent based on the lack of usable modularity constraints. Anyway, when p and q are similar magnitude roughly one in (n²/4) ln² b will both be prime this goes down(up?) to one in (n²-1) ln² b when they are at the other extremes of their ranges. The difference of (ln b)² can be usefully neglected when n is large enough to be interesting.

This means that we have (ln b)/2 ways of which only one in (n²/4) ln² b are acceptable. This cancels nicely to yield a density of 2-brilliants of one in (n²/2) ln b.

This gives excellent agreement with computed densities for instance the density of the first 1000 base 3 2-brilliants greater than 3⁵⁷ is one in 1765.118 and (57²/2) ln 3 = 1784.7

We can therefore expect that the density of 2-brilliants near 10⁹⁵ will be about one in 10390. The factor of ln b in the formula completely explains the density's base dependence left unexplained by the previous analysis.

From a handwavey point of view you can try and explain the base dependence by observing that 2⁹⁵+x is smaller than 10⁹⁵+x but then one has to mumble incoherently about being more likely to split into similar sized factors and I can't really justify it.

I have done a theoretical analysis of the density of 3-brilliants.
For numbers bⁿ±x the proportion of 3-brilliants is 1 in (n³ × ln b)/9 for n = 2 (mod 3) and 1 in 4 × (n³× ln b)/9 for n = 1 (mod 3).

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 September 25th, 2022.

Brilliant numbers

Table of contents

Maximal and minimal 2-brilliant numbers

Density of brilliant numbers (by Richard Heylen)