Largest 10^147-c Brilliant Number (p74*p74)
 

[QUOTE]10^147-28239 = p74 * p74,
10749227813857812842560638866039571941385856753480714536650423387382761653
93029938272478375299202950793413839971743799143685170784039618216410820237
[/QUOTE]

No 10^147-c with c>0 smaller than 28239 splits as p74*p74.

The attached file shows one factor (prime or composite but with less digits)
for any of these c's - or it indicates that 10^147-c is prime.

Since [url]https://www.alpertron.com.ar/BRILLIANT.HTM[/url] is missing this number,
I hope it is new.

Largest 10^151-c Brilliant Number (p76*p76)
 

[QUOTE]10^151 - 13731 = p76 * p76
2957232918876522618810099296027476287595656664821641247443138343849506540789
3381539525063545948219909981524129301893159430786891936664579050739840837321
[/QUOTE]

No 10^151-c with c>0 smaller than 13731 splits as p76*p76.

The attached file shows one factor (prime or composite but with less digits)
for any of these c's - or it indicates that 10^151-c is prime.

[url]https://www.alpertron.com.ar/BRILLIANT.HTM[/url] does not show this number yet.

Could I reserve 152 digits? I've already begun sieving and am about 10 SNFS factorizations in.

Yes, of course.
Thank you for your information.

I'd like to take the numbers 10^153-c.

Largest 10^153-c Brilliant Number (p77*p77)
 

[QUOTE]
10^153 - 24783 = p77 * p77
19616824731935528323413429323217946126629367330558378295828024144442130486423
50976649568165522929056351388822365973469595971165560920473974037361186978679
[/QUOTE]

No 10^153-c with c>0 smaller than 24783 splits as p77*p77.

The attached file shows the smallest prime factor
for any of these c's - or it indicates that 10^153-c is prime.

[url]https://www.alpertron.com.ar/BRILLIANT.HTM[/url] does not show this number yet.

10^151+14541 =
1892066823280306942402672399836621036921105169101702440417171708556479073929 *
5285225594021482704171643427873368811118569345423620874203438445689349531429

Factors of all n=10^151+c with c < 14541 and n either prime or with no prime factors < 20M attached in the file.

Also the entry for 35 on here is incorrectly listed on [url]https://www.alpertron.com.ar/BRILLIANT.HTM[/url], 10^35-783 = 290795768932439557 * 343883958033904381 is a larger 35 digit brilliant number.

Reserving 10^155+c (156 digits).

Largest 10^86 - c 3-Brilliant Number (p29 * p29 * p29)
 

[QUOTE]
10^86 - 2007389 splits as p29 * p29 * p29
33967693446721619162411558729
46163945373677107485245801167
63772144891696172285401983877
[/QUOTE]

10^86 - 2007389 is the largest 86-digit number with three 29-digit prime factors.

The appended file proves this.

Any line (except the last one) has exactly two entries.

The first shows c.

The second is either the letter 'p' (indicating that 10^86-c is prime) or
a prime factor of 10^86 - c with length different from 29.

The file contains a line for any odd 1 <= c <= 2007389, of course.

10^155+7213 is the smallest 156-digit number that factors into two 78-digit primes:

[QUOTE]
p78 factor: 211747224607852036333730181480768862317143398243227651081775505848031153848949
p78 factor: 472261207603529495511913284148545388735587409943720544148422568230650370558937
[/QUOTE]

Factor file for 10^155+c attached.

2^303 - 39727 is the largest 303-bit number that splits into two 152-bit primes:

[CODE]
p46 factor: 3755140210209107891033403488267039571419053517
p46 factor: 4339728185480567523635423762618623528383356693
[/CODE]

A few more 2-bit brilliant numbers:

[CODE]
2^297 - 7405 =
465449598594965125759568006862384515773478687 *
547061374229242156241055675491722178284755341

2^297 + 4301 =
447742596479348597082150804288878401042937171 *
568696163920948572072084351914463588415143263

2^299 - 31527 =
1006551460876452757803140363636308794862182167 *
1011888639335310739700404874667161763668291983
[/CODE]

Remaining base 2 brilliant numbers below 2^300:

[CODE]
2^295 - 13429 =
198510202319234021827282742860868097914815709 *
320675579978917834648228904658346016622753671

2^295 + 175343 =
199744320796412845995530912316777151860992621 *
318694288812019609071170106942006301252918091
[/CODE]

2^295 + c took an unusually long time with nearly 700 SNFS factorizations and 8 p45*p45 (147-bit * 149-bit) near misses before one was finally found:

[CODE]
17679 135424819131071650335816726305171046618468269 (45 digits)
28295 125056028398373651690810269127016407925857333 (45 digits)
68783 137665243947612845545922338816114028907278423 (45 digits)
90429 129289940650813391118621997356910828821798721 (45 digits)
112521 131584492955893608563695147526439257589143029 (45 digits)
124865 119468736668192803702619666384916214225482473 (45 digits)
148539 123275618234627848456962842180515532135679871 (45 digits)
165689 177963429546572797306584739591706914778068901 (45 digits)
[/CODE]

10^199-c
 

I'd like to continue with these numbers.
It will take a while.

Largest c199, splitting as p100 * q100
 

After a longer search, starting in September (!), I found this factorization:

[CODE]
10^199 - 5399 =
16656894422236107849246117001467083363915297020262\
62362095233433137957466266604897548197173600787241 *
60035200719352030115130596816375251478500171431803\
22684010592432041391073140089147545414675783616961
[/CODE]

Since c199 = 10^199 - 5399 is the largest 199-digit number with two 100-digit prime factors,
it is the 199-digit 2-brilliant number in base 10.

The attached file should help to check this assumption easily.
Any line in the file has two entries, the first is c, indicating the number,
the second is the smallest prime factor (of 10^199-c) or the letter 'p'.

10^157 - 2049 is brilliant
 

10^157 - 2049 is the largest 157-digit number with two 79-digit prime factors.

[CODE]
10^157 - 2049
1446814727739974723827555617027876206977909532017942734175294759323847644834853
6911735005366371832787385196708060058647586415356721193781047891582488320067667
[/CODE]

The format of the attached "proving file" did not change.

[QUOTE=Alfred;557937]No 10^147-c with c>0 smaller than 28239 splits as p74*p74.

The attached file shows one factor (prime or composite but with less digits)
for any of these c's - or it indicates that 10^147-c is prime.

Since [url]https://www.alpertron.com.ar/BRILLIANT.HTM[/url] is missing this number,
I hope it is new.[/QUOTE]

This site misses 154 digits.

[b][color=red]MODERATOR CAUTION: Please stop posting inane "observations" like this. Like the next poster, and many other users, I am sick of them.

I am sure the people running projects like this know about the lacunae in their tables.

If you want to do something constructive about a missing entry, fill it in or at least do something to forward that aim. Otherwise, please hold your peace.[/color][/b]

[QUOTE=sweety439;565081]This site misses 154 digits.[/QUOTE]

What is the worthy cause of this outstanding observation?

[QUOTE]
2^305-48019 = 6072848346102017942843738613347632788489697979 * 10733867787846864116326058114546589128269197047
[/QUOTE]

Apparently someone had already found this factor as the entry shows up on factordb.com and was created some time before November 2018. A google search of either factor gives nothing.

Sieving of 10^199+c continues, with no even splits found so far for c < 16000.

I'd like to take 10^201 - c.

10^91-1250741 is 3-brilliant
 

[CODE]
10^91 - 1250741
1130907742078987214643014936029
2064984395278432435681546498733
4282092540978287980470405630787[/CODE]

There is no larger 91-digit number which splits into three equally sized prime factors.

The attached file should prove this.

I omitted any line where 10^91-c is prime or where the smallest
prime factor (of 10^91-c) is smaller than 10^6.

PS: I'm still playing with 10^201-c.

[QUOTE=Alfred;567220]

The attached file should prove this.

[/QUOTE]

But it does not.
The file contains 282 lines, where the smallest primefactor has 31 digits.
For these numbers the attached file shows that the remaining cofactor is a 61-digit prime.

Now the proof should be complete.

I apologize the circumstances.

[QUOTE=swishzzz;566824]Apparently someone had already found this factor as the entry shows up on factordb.com and was created some time before November 2018. A google search of either factor gives nothing.[/QUOTE]

I played around a bit with creating a python script to find base-2 brilliant numbers and found several, but all of them were previously reported to factordb.com. I'm starting to suspect that there are even more, though I'm not sure how to scrape the database to find them without adding a lot of unfactored composites to the database. Perhaps someone on this forum has more experience with that, and would be willing to have a look?

[CODE]
2^311-9397
=
62370919584932696459960851277070596883408884321
*
66887737222666271935456033050473264106633618731

2^311+69711
=
51666465841110031879560999958408509880659269639
*
80745791522933342211102055412622336041844745081

2^313-24133
=
100788170265999753017323085706257528483989089343
*
165569021385057306060482886322484491911556750213

2^313+8505
=
118458567629160086527486150975362030803169102833
*
140871184348377129067049239080578375237738238409

2^315-19015
=
216834485254286594903496585433315946327767764161
*
307836619227101788652208469732562326355490574073

2^315+42701
=
228669422455046776001485409864826671274859838381
*
291904331396343474258089280492317954768687954849
[/CODE]

I have a Python web scraper that I use to check entries of the form n + c for varying c on factordb, though even filtering out composites which do not have a prime factor below 20M still generates about 3000 new factordb entries on a 100,000 search range for c. I suppose I can do some "light ecm" to further reduce that list by a factor of 5 or so, but it still takes a few hours to run a t25 on 3000+ 95-digit composites.

It would be nice if there was an easy way to find the smallest number that has been uploaded to factordb larger than a given number (as well as the largest number in factordb smaller than a given number), but I'm not sure how the numbers are stored internally in the database and doing such a query may be infeasible with the current infrastructure as there are over a billion entries in there already.

base 2 brilliant numbers
 

All found before November 2018.

[CODE]
2^305 + 47261 =
7316490771476709190807652965574102572379302277 *
8909346472058358236842543861359565227710116409

2^307 - 19027 =
11775430370240476555096243348888791129599911451 *
22142766486885472775913011741866569768555016151

2^307 + 371 =
14049124283454209175234576558264433714634147209 *
18559206944869223986131361958393223284004342811
[/CODE]

Adding 2^309±c to the above:

[CODE]
2^309-31899 =
24781986941524378779451719636879394253653915667 *
42085504376393744737222716439904864143412540839

2^309+19499 =
23517014357815368443003507125370993433302283221 *
44349270022733603968886915824049811627149524991
[/CODE]

Largest 10^n - c with exactly five equally sized prime factors
 

The solutions for some smaller n are as follows.

[CODE]
n c p q r s t
======================================================================
41 423906303 103948841 114710363 149953891 203482813 274847773
39 170690097 47957491 57090881 62357359 72085207 81253661
38 17945273 14859149 26589149 49138601 69387613 74232979
37 3737967 12265069 12316999 26021071 28269503 89987411
36 117140009 11983739 11997917 13976143 19075949 26087251
34 8846847 3448663 4321529 7868789 9038629 9434119
33 992687 1743487 3112091 5050681 5613833 6500093
32 6088257 1679213 1796947 2128781 2156351 7219523
31 5846847 1022701 1176899 1384829 2319659 2586377
29 13773393 303871 689957 709909 747157 899237
28 575559 231269 236563 445649 486821 842507
27 2939079 107837 133669 157349 468113 941861
26 13996773 104471 105361 150889 219017 274909
24 2962853 21961 65563 79309 92623 94547
23 3227433 10253 45673 46573 61729 74279
22 1218891 11807 15391 22189 35809 69257
21 3964821 10007 13537 16811 16921 25951
19 6735117 3041 4129 8581 9433 9839
18 343229 1583 2707 3691 6577 9613
17 416447 1327 1483 3001 3371 5023
16 1176773 1033 1229 1319 1447 4127
14 495633 181 751 859 859 997
13 50327 281 293 347 571 613
12 59901 131 131 211 277 997
11 423887 101 103 113 257 331
9 797979 41 61 67 67 89
8 4317 17 29 43 53 89
7 5411 11 17 19 29 97
[/CODE]

Lines for the smallest n are written for the sake of completeness only.

I was not able to compute the results for the larger n by an algorithm -
firstly caused by the very low level of my programming knowledge but
secondly maybe by the running time of a well written program?

I'd like to get a rough estimation.

PS: Continuing with 10^201-c

The Big Four
 

I've found the largest four 67-digit numbers with exactly four equally sized prime factors.

In 10^67 - c representation:

[CODE]
c p q r s
===========================================================================================
139852557 31431974044879763 46128143017681421 70125424881114271 98352831098894971
171158457 33992983359529783 34882916747820623 90179687845567937 93516705612649471
183261399 38966262167041379 40234767205193857 71686576788802147 88975803112152161
188505137 28896480880207309 43989844864448881 88028052195229997 89367893515033351
[/CODE]

The size of my proving file exceeds the 4.00 MB limit for .7z attachments by far.

PS: I do not forget about 10^201 - c.

base 2 brilliant numbers
 

[QUOTE]
2^317 - 3369 =
436065321852177727665353843557839743263823284547 *
612289870600221313399896732319435654464341654749
[/QUOTE]

A couple more found by Branger on 2020/12/30 which I don't have proof files for that are not listed on [url]https://www.alpertron.com.ar/BRILLIANT2.HTM#twobr:[/url]

[QUOTE]
2^313 - 24133 =
100788170265999753017323085706257528483989089343 *
165569021385057306060482886322484491911556750213

2^313 + 8505 =
118458567629160086527486150975362030803169102833 *
140871184348377129067049239080578375237738238409

2^315 - 19015 =
216834485254286594903496585433315946327767764161 *
307836619227101788652208469732562326355490574073

2^315 + 42701 =
228669422455046776001485409864826671274859838381 *
291904331396343474258089280492317954768687954849
[/QUOTE]

[QUOTE=swishzzz;572611]A couple more found by Branger on 2020/12/30 which I don't have proof files for that are not listed on [url]https://www.alpertron.com.ar/BRILLIANT2.HTM#twobr:[/url][/QUOTE]

It seems I forgot to post the proof files I had, thank you for the reminder.

Largest 10^n - c with exactly five equally sized prime factors, part 2
 

The solutions for n = 42, 43, 44 are as follows.

[CODE]
n c p q r s t
======================================================================
44 1561244849 182407289 762137381 766498163 960663391 976882183
43 15301041 104012933 175136021 748629481 781365433 938460631
42 46553637 108481379 192785767 207688121 301662217 763198063
[/CODE]

PS: 10^201-c is ongoing (with low priority).

I tried to look for brilliant numbers of the form 2^351+-c, and at least these ones were not in the factordb previously.

[CODE]

2^351-37629
=
54108213336623751930603114360343420697261006546122961
*
84774509249511338538855788282246318289861325315685779


2^351+74939
=
64719303637275716983586303735926609113831544522059491
*
70875256286568216854922693686888410014999992674845257

[/CODE]

Proof files are attached.

While doing aliquot factoring I came across this humble C114 = P57 * P57:

[CODE]140257568274260684468077810723210454538642881063207195453406733448965467421304070266933449577747915090059669116521
=
455667808354640170880989459219806232327469502542982654137
*
307806620750132280327670158666075502855210576269751690033[/CODE]
Yes, I know I could just have created this on the fly, but I didn't... ;)

Largest 10^n - c with exactly five equally sized prime factors, part 3
 

The solutions for n = 46, 47, 48, 49 are as follows.
[CODE]
n c p q r s t
======================================================================
49 292349261 4199063917 6046596823 6692964461 7518662791 7826676379
48 19443339 1039529507 4173506713 4220264131 6326926399 8632362059
47 84075503 1186228591 2527734631 2889263309 3333025919 3463174067
46 360258269 1055755553 1528954747 1668451013 1794793691 2068778527
[/CODE]

I'm in doubt about any extension of this table.

PS: I've downgraded 10^201-c to very low priority.

smallest 340 bit brilliant number
 

Test run of Amazon EC2 free tier. A 103 digit snfs job with factmsieve.py takes around 2.5 hours on a single t2 micro Windows instance running at 10% CPU capacity, perhaps this will be faster on a Linux instance with CADO.

[CODE]
2^339 + 15885

Sat Apr 24 15:56:40 2021 p51 factor: 887592350957138861091733941658539740396245192826267
Sat Apr 24 15:56:40 2021 p52 factor: 1261696734859514200896533536322632897894845904544119
[/CODE]

I made up some parameters and ran this C103 as a quartic SNFS job on CADO. Took a 2-hyperthreaded instance about 6 minutes to solve on a 12-core machine with 18 other threads busy.

And six more base-2 brilliant number, one of which took much longer than expected to find..

[CODE]
2^353-4203
=
125269909676882800262400105725282077739582106921689791
*
146467647140855997175976549181141923664201888942622379

2^353+20321
=
107212617965684231006696661224016764909923510204087201
*
171136469531909496395965898436670289930931414420683713

2^355-6237
=
201078246063474095502015538594857010445540999246436799
*
364992022501105116452215910637465538969268898994717469

2^355+11609
=
212926542994048279183611705107994489885983607534577991
*
344682042359245646786699688221203535450205665111934047

2^357-84009
=
453147361357734286103874959805167254938456672363539113
*
647841845458687143039115572227866832044273303789018751

2^357+293447
=
470652756196708146570444118328602847134424064755521003
*
623746103643411393320202884671105782953351092003850773
[/CODE]

None of these were in the factordb before at least, so they may be new.

[QUOTE=swishzzz;576780]Test run of Amazon EC2 free tier. A 103 digit snfs job with factmsieve.py takes around 2.5 hours on a single t2 micro Windows instance running at 10% CPU capacity, perhaps this will be faster on a Linux instance with CADO.

[CODE]
2^339 + 15885

Sat Apr 24 15:56:40 2021 p51 factor: 887592350957138861091733941658539740396245192826267
Sat Apr 24 15:56:40 2021 p52 factor: 1261696734859514200896533536322632897894845904544119
[/CODE][/QUOTE]

2^339 ± c completed:

[CODE]
2^339 - 27235 =
967306904908920452789078319450002924493095732788319 *
1157721882688666313356029082874219101278312435085187
[/CODE]

Four more base-2 brilliant numbers:

[CODE]
2^359-123577
=
818923988648227215894707705730371540540812518512510431
*
1433919762596343433061717035212209825301021813200029081

2^359+14621
=
1005099120748900459161798514928920510471822535126047147
*
1168313917648207456964919118126527398761685430631874647

2^361-6273
=
1560537277762292011030721646497096890825992876952316309
*
3009915387784249643267311161037214234010646089476820931

2^361+44571
=
1588267509671791256308972554753220446511580018944216381
*
2957364006343175263301783757664193291901358679995452983

[/CODE]

Proof files are attached.

And another batch finished:

[CODE]
2^363-291
=
4094013899900989030912375965276208433376145195035983021
*
4589222489607338458027850363512336418671632190977908977

2^363+163109
=
3594544799178917497808832006387639836453189974351828487
*
5226904020359514021042583817732061323376054356093003891

2^365-21055
=
6610188871706148247866877699982575091248185129696213497
*
11369321528836272916753163939935476417901514113750418441

2^365+25151
=
6975722917193176731536583299621002374191571619112667517
*
10773559033362830224264470417875672248024627915449550699
[/CODE]

Proof files are attached.

[QUOTE=Branger;591248]Proof files are attached.[/QUOTE]

Well, apparently not. Let's try that again.

Brilliant 201-digit number (10^201-c)
 

"Releasing" 10^201-c.
Tested any 1 <= c <= 40000.

Releasing 10^199+c as well. No p100 * p100 found from 0 < c < 100000.

After a lot of work by Eric Jeancolas, the table of minimal and maximal base-2 brilliant numbers is complete up to exponent 400, as you can see at [url]https://www.alpertron.com.ar/BRILLIANT2.HTM[/url].

Two more for the table:

[CODE]

10^181-28457
=
1976780995904575262824882988600377384132720583677328095507348810273221986844127586224818989
*
5058729328498020382937746413638967841933983271360558098096714142401565169191733122358525587

10^181+38439
=
1054300060000730056488062242818121661270786887706547586912142454641191605667380983978269333
*
9484965788575479583285807708150981397722931500035054082588713947753467817475117863857748683
[/CODE]

Proof files are attached.

I'm still working on 10^221+c, and I'm currently at about c=27000. It is slow going, however, so I will have to see how long my patience lasts.