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2021-04-24, 14:05   #1827
axn

Jun 2003

5,081 Posts

Quote:
 Originally Posted by charybdis Actually L and M. Jon has a detailed explanation on his page.
Thank you!

 2021-04-25, 10:12 #1828 bur     Aug 2020 4548 Posts 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?
2021-04-25, 13:32   #1829
charybdis

Apr 2020

5×73 Posts

Quote:
 Originally Posted by bur 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.

2021-04-26, 04:08   #1830
jyb

Aug 2005
Seattle, WA

176110 Posts

Quote:
 Originally Posted by bur 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?

 2021-05-18, 15:06 #1831 jyb     Aug 2005 Seattle, WA 3×587 Posts Time for an update 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
 2021-07-18, 16:04 #1832 jyb     Aug 2005 Seattle, WA 3·587 Posts Update 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
 2021-07-22, 05:35 #1833 jyb     Aug 2005 Seattle, WA 3·587 Posts 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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