Re: Pisot Numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg76412] Re: [mg76335] Pisot Numbers
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Mon, 21 May 2007 06:11:12 -0400 (EDT)
- References: <200705200621.CAA05765@smc.vnet.net> <B342ED46-0742-434D-9B26-60F80187246C@mimuw.edu.pl>
On 20 May 2007, at 18:03, Andrzej Kozlowski wrote: > > On 20 May 2007, at 15:21, dimitris wrote: > >> In view of a recent Message >> >> In[389]:= >> o1 = (E^(Sqrt[163]*Pi) - 744)^(1/3); >> o2 = 640320; >> >> In[391]:= >> N[(E^(Sqrt[163]*Pi) - 744)^(1/3) - 640320, 100] >> >> Out[391]= >> -6.096826476805298734971644609739663360781003906385187541696114645884 >> 6138003732168221294047257351725890381494173`100.*^-25 >> >> Also >> >> In[394]:= >> Element[o1, Integers] >> >> Out[394]= >> False >> >> Take now an example from Trot's Gidebooks. >> >> In[397]:= >> o3 = ((2/(27 + 3*Sqrt[69]))^(1/3) + (1/3)*((1/2)*(27 + >> 3*Sqrt[69]))^(1/3))^27369; >> >> o4 = >> 248872083860566242801488633985778816168566582615463984666186327177996 >> 8897\ >> 941302876969944745816129045615885143011927101923791713997993058914014 >> 883941331\ >> 496588665859636179886756365479484076315048561102041450220571014497428 >> 072837453\ >> 490447134892293461819188050968748780135755569233537426736962247783202 >> 459889540\ >> 213301883484666470466149889402655143734621040204402439497074243583844 >> 435180857\ >> 228403580970629296798899333826598686243987854716724374760335810100582 >> 327703252\ >> 886711404982379820790899904312876809580414490656116484737937974600066 >> 542685289\ >> 106532890742345783983687027507936729079442473934078360160815378816949 >> 415366223\ >> 547953896457883387197030107324924232558604649327195920807344164169408 >> 849950012\ >> 979654395273385341095562256314722477722302818244400186545582913013684 >> 116069229\ >> 948450508385560502376379491505913877574694543067098950233734987525958 >> 694493166\ >> 065786146114295805170616134580156268741967789244572258673205513485511 >> 448982113\ >> 074128616447024942770432196754923847050903086833932583983456210775092 >> 840495926\ >> 289398412204946622896060874294857076651762085967637510807753767056601 >> 346018771\ >> 027068086233850837047631634161338416471812349025685230145549063307448 >> 984654469\ >> 500345708114334002372857024261410333404070216793731889901563587912181 >> 986503488\ >> 932240588333472792264516219643268144193209629883467045872736189979709 >> 366330108\ >> 944683622923025480388609270892579905058376065654372722673382421099596 >> 652032752\ >> 423655970286505879088423573116299843248723992370681856106228825253081 >> 951335763\ >> 606805097314767756009989894248180226689216688712554660307978677642033 >> 917433524\ >> 177703612346235567428057168862868263715487449187865230223959037178478 >> 650607885\ >> 929852524030200605375426361295649137497579902728693786036767203892699 >> 418847034\ >> 973900792486513050707875184722293046768355234117849762278847536427384 >> 240325375\ >> 931710068928003083282083508258941657571106418546338991654633520007125 >> 094003937\ >> 060577513244349419124583678640310438044741715469307650984762987113625 >> 655095113\ >> 341410659514797573216487308588020792972361604798011836953448415069774 >> 170327604\ >> 176428382899037366367969875803830362244613565592323446457417387836546 >> 707590791\ >> 148857442335097804365308142758237796222541372347526347511124157083242 >> 577253654\ >> 864546653468558226069365021560451385770280243507694206247762400972408 >> 775051143\ >> 528825344094380032368218145009068738988932699440006161647412432021399 >> 929998924\ >> 197063449517037778261055705878691043258271291941546764790768702904202 >> 815388755\ >> 953467402952252786624210537218217362187375224335225100774863989100606 >> 085031055\ >> 987180950433574684009505526256479756716140052888061921437953507269705 >> 531834507\ >> 752244853777872848075149669430514248120843405866305425664958833381695 >> 289311873\ >> 275612903811562531683996339721232710796969624597692084825522259134899 >> 944567445\ >> 316144180191492624723899611977533345482296723851296876182987982763612 >> 903081830\ >> 406428255761789360866674785134042824865250319983289744838888137526494 >> 195021927\ >> 158720998042457987098509876243983825524393130319382015891243101298654 >> 993872084\ >> 034846505853704619531981994143584471102830065857739428507878016585984 >> 828808526\ >> 342887038330953482823346606566055339838200632031259942468414620516606 >> 902878898\ >> 295905037327168661392320861496592384492793915926275510204303513646878 >> 274710211\ >> 927798593011178010654392195694992994203684249930039904616401126153259 >> 826319808\ >> 971152916585811064172283699654029309129460623214205826005262694547534 >> 088; >> >> o3 is not an integer, but it nearly is. >> >> In[401]:= >> N[(2^(1/3)/(27 + 3*Sqrt[69])^(1/3) + (27 + 3*Sqrt[69])^(1/3)/ >> (3*2^(1/3)))^27369, 5030] - >> 248872083860566242801488633985778816168\ >> 566582615463984666186327177996889794130287696994474581612904561588514 >> 30119271019237917139979930589140148839413314965886658596361\ >> 798867563654794840763150485611020414502205710144974280728374534904471 >> 34892293461819188050968748780135755569233537426736962247783\ >> 202459889540213301883484666470466149889402655143734621040204402439497 >> 07424358384443518085722840358097062929679889933382659868624\ >> 398785471672437476033581010058232770325288671140498237982079089990431 >> 28768095804144906561164847379379746000665426852891065328907\ >> 423457839836870275079367290794424739340783601608153788169494153662235 >> 47953896457883387197030107324924232558604649327195920807344\ >> 164169408849950012979654395273385341095562256314722477722302818244400 >> 18654558291301368411606922994845050838556050237637949150591\ >> 387757469454306709895023373498752595869449316606578614611429580517061 >> 61345801562687419677892445722586732055134855114489821130741\ >> 286164470249427704321967549238470509030868339325839834562107750928404 >> 95926289398412204946622896060874294857076651762085967637510\ >> 807753767056601346018771027068086233850837047631634161338416471812349 >> 02568523014554906330744898465446950034570811433400237285702\ >> 426141033340407021679373188990156358791218198650348893224058833347279 >> 22645162196432681441932096298834670458727361899797093663301\ >> 089446836229230254803886092708925799050583760656543727226733824210995 >> 96652032752423655970286505879088423573116299843248723992370\ >> 681856106228825253081951335763606805097314767756009989894248180226689 >> 21668871255466030797867764203391743352417770361234623556742\ >> 805716886286826371548744918786523022395903717847865060788592985252403 >> 02006053754263612956491374975799027286937860367672038926994\ >> 188470349739007924865130507078751847222930467683552341178497622788475 >> 36427384240325375931710068928003083282083508258941657571106\ >> 418546338991654633520007125094003937060577513244349419124583678640310 >> 43804474171546930765098476298711362565509511334141065951479\ >> 757321648730858802079297236160479801183695344841506977417032760417642 >> 83828990373663679698758038303622446135655923234464574173878\ >> 365467075907911488574423350978043653081427582377962225413723475263475 >> 11124157083242577253654864546653468558226069365021560451385\ >> 770280243507694206247762400972408775051143528825344094380032368218145 >> 00906873898893269944000616164741243202139992999892419706344\ >> 951703777826105570587869104325827129194154676479076870290420281538875 >> 59534674029522527866242105372182173621873752243352251007748\ >> 639891006060850310559871809504335746840095055262564797567161400528880 >> 61921437953507269705531834507752244853777872848075149669430\ >> 514248120843405866305425664958833381695289311873275612903811562531683 >> 99633972123271079696962459769208482552225913489994456744531\ >> 614418019149262472389961197753334548229672385129687618298798276361290 >> 30818304064282557617893608666747851340428248652503199832897\ >> 448388881375264941950219271587209980424579870985098762439838255243931 >> 30319382015891243101298654993872084034846505853704619531981\ >> 994143584471102830065857739428507878016585984828808526342887038330953 >> 48282334660656605533983820063203125994246841462051660690287\ >> 889829590503732716866139232086149659238449279391592627551020430351364 >> 68782747102119277985930111780106543921956949929942036842499\ >> 300399046164011261532598263198089711529165858110641722836996540293091 >> 29460623214205826005262694547534088 >> >> Out[401]= >> -0.999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 999999999999999999999999999999999999999999999999999999999999999999999 >> 999999999\ >> 99999999999999999999999999999999999881537021130425452 >> >> >> However Element[o3,Integers] return unevaluated. >> >> In[404]:= >> Element[o3, Integers] >> >> Out[404]= >> ((2/(27 + 3*Sqrt[69]))^(1/3) + (1/3)*((1/2)*(27 + >> 3*Sqrt[69]))^(1/3))^27369 Integers >> >> Why? >> How we can symbolically show that o3 is not an integer? >> >> Dimitris >> >> > > > But it is extremly easy! > > o3 = ((2/(27 + 3*Sqrt[69]))^(1/3) + (1/ > 3)*((1/2)*(27 + 3*Sqrt[69]))^(1/3))^27369; > FullSimplify[Element[o3, Integers]] > False > > ?? > > Andrzej Kozlowski I should add that using FullSimplify does not always mean that we are proving something "symbolically", because increasingly in computer algebra algebraic functions rely on 'symbolically verified' numerical methods, that is, numerical methods that return answers that are as valid as those returned by purely symbolic methods. Such methods, when they exist, tend to be a lot faster than purely symbolic ones. So in this case it is also possible that this kind of technique is used by FullSimplify. I can only guess of course... Andrzej Kozlowski
- References:
- Pisot Numbers
- From: dimitris <dimmechan@yahoo.com>
- Pisot Numbers