Re: Pisot Numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg76411] Re: [mg76335] Pisot Numbers
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 21 May 2007 06:10:41 -0400 (EDT)
• References: <200705200621.CAA05765@smc.vnet.net>

```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]=
> -6.0968264768052987349716446097396633607810039063851875416961146458846
> 138003732168221294047257351725890381494173`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 =
> 2488720838605662428014886339857788161685665826154639846661863271779968
> 897\
> 9413028769699447458161290456158851430119271019237917139979930589140148
> 83941331\
> 4965886658596361798867563654794840763150485611020414502205710144974280
> 72837453\
> 4904471348922934618191880509687487801357555692335374267369622477832024
> 59889540\
> 2133018834846664704661498894026551437346210402044024394970742435838444
> 35180857\
> 2284035809706292967988993338265986862439878547167243747603358101005823
> 27703252\
> 8867114049823798207908999043128768095804144906561164847379379746000665
> 42685289\
> 1065328907423457839836870275079367290794424739340783601608153788169494
> 15366223\
> 5479538964578833871970301073249242325586046493271959208073441641694088
> 49950012\
> 9796543952733853410955622563147224777223028182444001865455829130136841
> 16069229\
> 9484505083855605023763794915059138775746945430670989502337349875259586
> 94493166\
> 0657861461142958051706161345801562687419677892445722586732055134855114
> 48982113\
> 0741286164470249427704321967549238470509030868339325839834562107750928
> 40495926\
> 2893984122049466228960608742948570766517620859676375108077537670566013
> 46018771\
> 0270680862338508370476316341613384164718123490256852301455490633074489
> 84654469\
> 5003457081143340023728570242614103334040702167937318899015635879121819
> 86503488\
> 9322405883334727922645162196432681441932096298834670458727361899797093
> 66330108\
> 9446836229230254803886092708925799050583760656543727226733824210995966
> 52032752\
> 4236559702865058790884235731162998432487239923706818561062288252530819
> 51335763\
> 6068050973147677560099898942481802266892166887125546603079786776420339
> 17433524\
> 1777036123462355674280571688628682637154874491878652302239590371784786
> 50607885\
> 9298525240302006053754263612956491374975799027286937860367672038926994
> 18847034\
> 9739007924865130507078751847222930467683552341178497622788475364273842
> 40325375\
> 9317100689280030832820835082589416575711064185463389916546335200071250
> 94003937\
> 0605775132443494191245836786403104380447417154693076509847629871136256
> 55095113\
> 3414106595147975732164873085880207929723616047980118369534484150697741
> 70327604\
> 1764283828990373663679698758038303622446135655923234464574173878365467
> 07590791\
> 1488574423350978043653081427582377962225413723475263475111241570832425
> 77253654\
> 8645466534685582260693650215604513857702802435076942062477624009724087
> 75051143\
> 5288253440943800323682181450090687389889326994400061616474124320213999
> 29998924\
> 1970634495170377782610557058786910432582712919415467647907687029042028
> 15388755\
> 9534674029522527866242105372182173621873752243352251007748639891006060
> 85031055\
> 9871809504335746840095055262564797567161400528880619214379535072697055
> 31834507\
> 7522448537778728480751496694305142481208434058663054256649588333816952
> 89311873\
> 2756129038115625316839963397212327107969696245976920848255222591348999
> 44567445\
> 3161441801914926247238996119775333454822967238512968761829879827636129
> 03081830\
> 4064282557617893608666747851340428248652503199832897448388881375264941
> 95021927\
> 1587209980424579870985098762439838255243931303193820158912431012986549
> 93872084\
> 0348465058537046195319819941435844711028300658577394285078780165859848
> 28808526\
> 3428870383309534828233466065660553398382006320312599424684146205166069
> 02878898\
> 2959050373271686613923208614965923844927939159262755102043035136468782
> 74710211\
> 9277985930111780106543921956949929942036842499300399046164011261532598
> 26319808\
> 9711529165858110641722836996540293091294606232142058260052626945475340
> 88;
>
> 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\
> 5665826154639846661863271779968897941302876969944745816129045615885143
> 0119271019237917139979930589140148839413314965886658596361\
> 7988675636547948407631504856110204145022057101449742807283745349044713
> 4892293461819188050968748780135755569233537426736962247783\
> 2024598895402133018834846664704661498894026551437346210402044024394970
> 7424358384443518085722840358097062929679889933382659868624\
> 3987854716724374760335810100582327703252886711404982379820790899904312
> 8768095804144906561164847379379746000665426852891065328907\
> 4234578398368702750793672907944247393407836016081537881694941536622354
> 7953896457883387197030107324924232558604649327195920807344\
> 1641694088499500129796543952733853410955622563147224777223028182444001
> 8654558291301368411606922994845050838556050237637949150591\
> 3877574694543067098950233734987525958694493166065786146114295805170616
> 1345801562687419677892445722586732055134855114489821130741\
> 2861644702494277043219675492384705090308683393258398345621077509284049
> 5926289398412204946622896060874294857076651762085967637510\
> 8077537670566013460187710270680862338508370476316341613384164718123490
> 2568523014554906330744898465446950034570811433400237285702\
> 4261410333404070216793731889901563587912181986503488932240588333472792
> 2645162196432681441932096298834670458727361899797093663301\
> 0894468362292302548038860927089257990505837606565437272267338242109959
> 6652032752423655970286505879088423573116299843248723992370\
> 6818561062288252530819513357636068050973147677560099898942481802266892
> 1668871255466030797867764203391743352417770361234623556742\
> 8057168862868263715487449187865230223959037178478650607885929852524030
> 2006053754263612956491374975799027286937860367672038926994\
> 1884703497390079248651305070787518472229304676835523411784976227884753
> 6427384240325375931710068928003083282083508258941657571106\
> 4185463389916546335200071250940039370605775132443494191245836786403104
> 3804474171546930765098476298711362565509511334141065951479\
> 7573216487308588020792972361604798011836953448415069774170327604176428
> 3828990373663679698758038303622446135655923234464574173878\
> 3654670759079114885744233509780436530814275823779622254137234752634751
> 1124157083242577253654864546653468558226069365021560451385\
> 7702802435076942062477624009724087750511435288253440943800323682181450
> 0906873898893269944000616164741243202139992999892419706344\
> 9517037778261055705878691043258271291941546764790768702904202815388755
> 9534674029522527866242105372182173621873752243352251007748\
> 6398910060608503105598718095043357468400950552625647975671614005288806
> 1921437953507269705531834507752244853777872848075149669430\
> 5142481208434058663054256649588333816952893118732756129038115625316839
> 9633972123271079696962459769208482552225913489994456744531\
> 6144180191492624723899611977533345482296723851296876182987982763612903
> 0818304064282557617893608666747851340428248652503199832897\
> 4483888813752649419502192715872099804245798709850987624398382552439313
> 0319382015891243101298654993872084034846505853704619531981\
> 9941435844711028300658577394285078780165859848288085263428870383309534
> 8282334660656605533983820063203125994246841462051660690287\
> 8898295905037327168661392320861496592384492793915926275510204303513646
> 8782747102119277985930111780106543921956949929942036842499\
> 3003990461640112615325982631980897115291658581106417228369965402930912
> 9460623214205826005262694547534088
>
> Out[401]=
> -0.9999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 9999999999999999999999999999999999999999999999999999999999999999999999
> 99999999\
> 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

```

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