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2021-04-24, 14:05   #1827
axn

Jun 2003

143B16 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 79*6581e-4;3*2539e-3 2·11·19 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

10000111012 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

6F616 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 110111101102 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 110111101102 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 2×34×11 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.
 2021-09-16, 04:54 #1834 jyb     Aug 2005 Seattle, WA 6F616 Posts Update I have updated the tables and ECMnet server to reflect all factors reported to me through the end of September 15 (GMT). There were 28 new factors found since the last update, of which 25 completed the factorization of their respective numbers. There are now 750 composites remaining in the tables. The range of SNFS difficulties remains as it was (233 to 310). All of the lowest-difficulty numbers are obligatory quartics, with the lowest non-quartic difficulty now at 259. There are a number of octics, the easiest of which has a nominal difficulty of 240. The minimum digit count has risen to 188, as several GNFS jobs have been completed. I have no information on two of the factorizations which were completed. If anybody knows who factored 7-2,311 or 7-6,349, please let me know. Here are the factors found since the last update, showing appropriate credit: Code: 9-5 665L C187 3860972841270514452233691632976217617100269677822059121399063498534992203592191551. P106 NFS@Home & J Becker GNFS 2021-07-18 10-9 263 C193 707620156731045648096326922095406943876636712533143444995325819391037. P124 NFS@Home & M Vang SNFS 2021-07-18 9-5 269 C213 238198368609970329041819957185968002298074564429218115171. P156 NFS@Home & S Wellman SNFS 2021-07-20 7-2 311 C256 1233818785651197615658160585019581693982901513874510799187. P199 Unknown UNK 2021-07-21 7-6 349 C243 20593914389069128477014389955174277353056954536392972073975705559. P178 Unknown UNK 2021-07-21 10+7 257 C212 13167659309807933661947865065167726298989732975339458377263084783. P148 NFS@Home & J Becker SNFS 2021-07-28 4+3 433 C241 48945722146142908103060580301841166905710187535747129950261795535742027. P170 NFS@Home & J Becker SNFS 2021-07-29 5+2 403 C185 475000477685691181017641158143829995323995188966193591406612384565594305901037. P108 NFS@Home & S Wellman SNFS 2021-07-30 8+7 658M C212 656272964586190282066864481688848890075330056596263373965940451584658077069100619277. P129 NFS@Home & S Wellman SNFS 2021-07-30 4+3 473 C207 11951958331057350155734401249878898754946783106272991980673217257959900838407758627573411032577. P113 NFS@Home & J Becker SNFS 2021-08-01 5+2 407 C200 84543959237764757287910084815693252637252308941378532019252576355629261214377161287238895327. P108 NFS@Home & J Becker SNFS 2021-08-08 4-3 437 C198 85957790811973669372192235983740670496852102766763398661674231571453. P130 NFS@Home & M Vang SNFS 2021-08-15 5+3 379 C195 1213752316618349925582586438775629204714780110113087155656930513564490147035659322720661. P108 NFS@Home & M Vang SNFS 2021-08-16 10+3 257 C232 3901262736896630274577345434447283792027017852795336316625809. C171 NFS@Home & S Wellman SNFS 2021-08-17 10+3 257 C171 964208386518575683272213770865535978093187727400955637689661050307516730729. P96 NFS@Home & S Wellman SNFS 2021-08-17 11+2 248 C200 119955541478842916612823882564717952056877375932044007782207518948823041. P129 NFS@Home & S Wellman SNFS 2021-08-18 5+3 407 C210 1740030651925052000148050106460781935520058747090580313091265861699. C144 NFS@Home & M Vang SNFS 2021-08-18 5+3 407 C144 89175774327527513051484791201473140752355783065152979750805068820139. P76 NFS@Home & M Vang SNFS 2021-08-18 10-3 257 C240 12903223717239310701599417516919708176943342363241303565526878670887031182547617953116289247055152318787487. P134 NFS@Home & J Becker SNFS 2021-08-24 6+5 331 C233 7107065894537645636474266095580536144484110194983012435157939385869339796319285140137. P148 NFS@Home & S Wellman SNFS 2021-08-26 7+2 314 C195 3554210610846605535575054210842480189141052368865103912262116620343051204265241487704759542760053. P98 NFS@Home & M Vang SNFS 2021-08-27 3+2 556 C186 517954206328921374428966056764161730583724062305371627573373160200787481. P114 NFS@Home & M Vang SNFS 2021-08-29 9-5 281 C195 70843838365619567462810228749785276065270991985155192411049058849. P130 NFS@Home & M Vang SNFS 2021-09-08 5+3 368 C242 70347627820233937826140058314439852420876393014659975914839637277297269952169860325083269746852410090843519993409729. P127 NFS@Home & S Wellman SNFS 2021-09-08 9+8 271 C254 57016320591952120412424461448678234035975146527927731496100496326020182547347261535675163229732744056224996337. P144 NFS@Home & E Engler SNFS 2021-09-09 3+2 551 C187 276289231332731566044884175074973665891294875814153042737. P131 NFS@Home & G Childers SNFS 2021-09-12 6+5 1170M C193 84425495929288974558620131958491524433468074153471662920261. C134 Yoyo@home ECM 2021-09-15 6+5 1170M C134 454485417305472722732273177555583822428174921499261067478866561. P72 Yoyo@home ECM 2021-09-15 Last fiddled with by jyb on 2021-09-16 at 04:57
 2021-10-05, 18:22 #1835 jyb     Aug 2005 Seattle, WA 2×34×11 Posts Missed report Whoops! Looks like I completely failed to notice RichD's factor report for 9-5,271, way back on Sept 12. It should have been in the last update. It will appear in the next one. Sorry, Rich! Last fiddled with by jyb on 2021-10-05 at 18:22

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