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Old 2021-04-24, 14:05   #1827
axn
 
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Quote:
Originally Posted by charybdis View Post
Actually L and M.

Jon has a detailed explanation on his page.
Thank you!
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Old 2021-04-25, 10:12   #1828
bur
 
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Thanks, I still don't understand why in jyb's list there are pairs such as 5+2,830M that refer to the Aurefeullian factorization, while there are other such as 4+3,424 where neither L nor M is shown. Why is that?


Aren't both just shorthand for a^n+b^n?
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Old 2021-04-25, 13:32   #1829
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Quote:
Originally Posted by bur View Post
Thanks, I still don't understand why in jyb's list there are pairs such as 5+2,830M that refer to the Aurefeullian factorization, while there are other such as 4+3,424 where neither L nor M is shown. Why is that?


Aren't both just shorthand for a^n+b^n?
Most a^n+b^n do not have an Aurifeuillian factorization. Jon's table of identities tells us that 4^n+3^n only has one when n is 3 times an odd number.
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Old 2021-04-26, 04:08   #1830
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Quote:
Originally Posted by bur View Post
Thanks, I still don't understand why in jyb's list there are pairs such as 5+2,830M that refer to the Aurefeullian factorization, while there are other such as 4+3,424 where neither L nor M is shown. Why is that?


Aren't both just shorthand for a^n+b^n?
No, they're not just shorthand for a^n+b^n. Did you read the page to which Robert directed you? It should be able to clarify, but if there's anything on it you don't understand, feel free to ask here (or via DM if you prefer) and we can try to help.

Of particular note is this part:
Quote:
If we let r be the squarefree part of a × b, then we have the following:
If r ≡ 1 (mod 4), then the Aurifeuillian factorizations will be "on the minus side." I.e. they will be found in the numbers of the form an - bn.
If r ≡ 2 or 3 (mod 4), then the Aurifeuillian factorizations will be "on the plus side." I.e. they will be found in the numbers of the form an + bn.
Numbers on the appropriate side have an Aurifeuillian factorization when their exponent is an odd multiple of r.
So in the case of the 4+3 table, only those exponents which are an odd multiple of 3 will give an Aurifeuillian factorization (as Robert already pointed out), thereby excluding 4+3,424. And in the case of the 5+2 table, only exponents which are an odd multiple of 10 will have such a factorization, so 5+2,830 qualifies (and therefore gives distinct 5+2,830L and 5+2,830M entries in the table), while 5+2,427 does not.

Does that help?
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Old 2021-05-18, 15:06   #1831
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The tables and ECMnet server have been updated to reflect all factors reported to me through the end of May 17 (GMT). There were 27 new factors found since the last update, of which 25 completed the factorization of their respective numbers.

There are now 800 composites remaining in the tables. With the factorization of 9+8,538L, the minimum digit count has now jumped to 185. With the usual polynomial degrees, SNFS difficulties still range from 233 to 310, though all difficulties below 255 are for quartics.

Here are the factors found since the last update, showing appropriate credit:
Code:
10-9	301	C247	1166050058398648290381182153579501524867943211569524161665693673407438005097477829470831046604052564300310697233981323923.	P127	NFS@Home & J Becker	SNFS	2021-02-26
8+3	834M	C220	161289010875891408149774896723759782914182820797075070649700914143222073547620631141.	P137			NFS@Home & J Becker	SNFS	2021-02-28
8+3	774L	C210	2124525938024025624819821285378386853416542355268093661987757793243606113.	P138				NFS@Home & J Becker	SNFS	2021-03-01
9+2	272	C185	2653541588463188643459641832202685148767413388720300427974220503215900801.	P112				NFS@Home & J Becker	SNFS	2021-03-03
3+2	1494L	C186	138496499736965293417784228061104580201912383277093096923971003462621.	P118					NFS@Home & J Becker	SNFS	2021-03-06
8+3	774M	C223	394919403261007891106577866024599716542953650239284344299419901724499859466699310235306392453.	P131		NFS@Home & R Smith	SNFS	2021-03-07
7+6	307	C183	4322962132266970946846187918408926764510073310299857275447117097708551341.	P110				NFS@Home & R Dickerson	SNFS	2021-03-14
11+7	244	C250	1059541973080742310555353784490717846000700811635333436715773635704846212282448747946836343381030372172465563706929.	P136	NFS@Home & J Becker	SNFS	2021-03-15
7+5	346	C186	45181777382655381933066327843180466474842349406427999710501275601976792932053935732142434857.	P94		NFS@Home & M Vang	GNFS	2021-03-17
8+7	602M	C227	4916917166368996931380374108874416747677560411659567907186930453841.	C160					NFS@Home & D Domanov	SNFS	2021-03-18
8+7	602M	C160	7884018211102719537360660116398527924151791252852261975686457049561.	P93					NFS@Home & D Domanov	SNFS	2021-03-18
4-3	511	C184	11902063434118475264317905358200965698872683674279936012424813992146338444334779713.	P102			NFS@Home & J Becker	SNFS	2021-03-19
11+3	251	C185	46170806290961721671472854912376407942290243052674602322562646748229921561739119774929.	P100			NFS@Home & J Becker	SNFS	2021-03-23
5-3	367	C200	985816600114738579524735301598577943592370169052614234608050594023899023287471667084838782083.	P107		NFS@Home & J Becker	SNFS	2021-03-25
5-3	407	C252	263425170884710138945113328196987857399367382281923172038554119.	C189					NFS@Home & J Becker	SNFS	2021-04-02
5-3	407	C189	1533141756059930858338678599627715718898784539943974746727366630018138994609.	P114				NFS@Home & J Becker	SNFS	2021-04-02
9-7	299	C198	135223516103616176320374158278958681368396873920312069963293467041443.	P130					NFS@Home & J Becker	SNFS	2021-04-03
5+4	367	C207	4742868738686077090058189821618954819892053024407909703490460726219926948901546229.	P125			NFS@Home & J Becker	SNFS	2021-04-11
12-5	239	C222	411580637283005215946023556288045458275149969844695794423538799829.	P157					NFS@Home & J Becker	SNFS	2021-04-18
10+7	296	C206	5195383704460532465905323233328652683649594898990851338811951409.	P142					J Becker		ECM	2021-04-21
8+7	316	C263	32823879912258735727626372991309710431020327877147706439666534226483113.	P193				J Becker		ECM	2021-04-23
7-5	347	C248	4118808381646490858492122710109296759660629242671559358457283.	P187						S Wellman		ECM	2021-04-28
9+8	538L	C180	338430633770856762857851157159040038937931454134294153004927655481823091137.	P106				NFS@Home & R Dickerson	SNFS	2021-05-03
3+2	1566L 	C233	535369722225843247953245064084415987151143374254614367936806164027658509852566093337695242313814360143767381.	P125	NFS@Home & J Becker	SNFS	2021-05-08
9+2	566L	C185	26179983493988917766980136640719912771239224393266767771232409068071046199586335693317.	P99			NFS@Home & M Vang	GNFS	2021-05-09
7+4	341	C189	3231774010815260361375943864820533820319254996863885567577810487150181.	P119					NFS@Home & J Becker	SNFS	2021-05-12
9+8	538M	C256	23493468309403547274284204755629964487513760629928865035150587896315990825986607587017872611047702971052874886539640708577.	P135	NFS@Home & D Domanov	SNFS	2021-05-17
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Old 2021-07-18, 16:04   #1832
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The tables and ECMnet server have once again been updated to reflect all factors reported to me through the end of July 17 (GMT). There were 27 new factors found since the last update, of which 25 completed the factorization of their respective numbers.

There are now 775 composites remaining in the tables. The minimum digit count remains 185, though several composites with length in the 180's have been factored. With the usual polynomial degrees, SNFS difficulties still range from 233 to 310; difficulties below 257 are for quartics.

Here are the factors found since the last update, showing appropriate credit:
Code:
11+5	251	C243	2431111417510092886627866877154305717347222560597974156970779.	P182						NFS@Home & J Becker	SNFS	2021-05-21
7+6	328	C186	1945006717945075909827104617083050955402888436945247382872729729.	P123					NFS@Home & J Becker	GNFS	2021-05-27
11+7	251	C190	580558091022345529674853359429535040006877128994971030819233360765231467869449147193.	P106			NFS@Home & J Becker	SNFS	2021-05-28
5-4	367	C248	20776725028772474229812530495883369039638610475295877643763432557797619722141.	P172				NFS@Home & S Wellman	SNFS	2021-05-29
8+3	284	C199	6985921953532407395841806973633655186174312451008572457777.	P141						NFS@Home & S Wellman	SNFS	2021-05-29
11+2	251	C198	327097982698738024087780616906671055097103068547871134004301537286675767592309.	P120				NFS@Home & J Becker	SNFS	2021-05-30
6+5	1110L	C186	9832533088063097790396449853168071102239580625095331899153432859661.	P119					NFS@Home & J Becker	SNFS	2021-06-06
5+3	1275L	C213	15485392175571482288572694463766962208189345454263471351.	P158						J Becker		ECM	2021-06-09
3-2	599	C286	57098442495458904828938842884105123315746212344766605767.	C231						J Becker		ECM	2021-06-09
11+4	284	C186	11130182168192502138420902758142553149791549728509225659937354330602882256322032257.	P104			NFS@Home & M Vang	GNFS	2021-06-11
11+10	248	C185	551424278037922916147804083650476833524216077674603541307766710961.	P119					NFS@Home & S Wellman	SNFS	2021-06-13
10+9	281	C187	7609626239410410804130723520322963312381066767260338386649177390894037.	P117					NFS@Home & J Becker	GNFS	2021-06-14
3+2	1758L	C190	20056257140525238827944708795427289880876895721011953040327799961.	P125					NFS@Home & J Becker	GNFS	2021-06-17
4+3	1533M	C208	1612630970354338886718929665345137865208737134366174655714641571.	P145					NFS@Home & J Becker	SNFS	2021-06-20
8+7	284	C197	23715493376640385699572145623803541286366621909134809307875312574238334224087723361561.	P111			NFS@Home & S Wellman	SNFS	2021-06-23
10+9	262	C195	8226379582950375868532044982744304838235786090349888873060428859620550702935742281.	P113			NFS@Home & M Vang	SNFS	2021-06-25
9+8	269	C221	482111404714170540930757635916611564311026442042593947491.	C164						NFS@Home & J Becker	SNFS	2021-06-30
9+8	269	C164	12013935185576168708506787885405767215243611179840019291971961093386049.	P94				NFS@Home & J Becker	SNFS	2021-06-30
7+6	304	C194	10435777068338092346047968104390762223738438380565704539847480139673403403441974486999843009.	P103		NFS@Home & S Wellman	SNFS	2021-07-01
10+3	930M	C201	15046653383457144998203878529204971588447503074491299672027844466646340867811603563168661.	P113		E Hall			SNFS	2021-07-03
8+3	846M	C242	98880975695090219597943319438209646049906218765931369611604267657612189452032332629236575446980772632081994457192101.	P126	NFS@Home & J Becker	SNFS	2021-07-04
4+3	1341M	C186	265966590821958160212473178851892352131547927210303756886145259.	P124					NFS@Home & M Vang	GNFS	2021-07-07
8+3	846L	C239	28992957901010279213750157502206064459999240076165878055519076804798670329755507636197580755909593298704622602337.	P127	NFS@Home & S Wellman	SNFS	2021-07-09
11+5	248	C217	3664054391198399558370179100559871921272450778962158214135669282431117515425751090017.	P133			NFS@Home & S Wellman	SNFS	2021-07-10
7+5	875L	C188	98342280365856308760847226402406620007342227815530637963975723845349742457574109759125001.	P99		NFS@Home & J Becker	GNFS	2021-07-11
7+3	347	C188	2325904027834780254606286969969619185459783229630940091062912401562650402721671.	P109			NFS@Home & M Vang	GNFS	2021-07-14
8-7	323	C188	19424029423642645153341244605275556654918591697507539417515766581943432434921047293397.	P103			NFS@Home & S Wellman	GNFS	2021-07-15
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Old 2021-07-22, 05:35   #1833
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Does anybody wish to claim credit for completing the factorization of 7-2,311? Its factors were entered into the FactorDB on 2021-07-21, at 11:44 GMT. I suspect it was done by ECM, but SNFS is plausible as well.
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