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Re: Pisot Numbers


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


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