Re: Pisot Numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg76409] Re: [mg76335] Pisot Numbers
• Date: Mon, 21 May 2007 06:09:38 -0400 (EDT)
• References: <200705200621.CAA05765@smc.vnet.net> <B342ED46-0742-434D-9B26-60F80187246C@mimuw.edu.pl> <CA3DA319-0683-4B8B-83D0-EFEDE5BC79C0@mimuw.edu.pl>

```Andrzej Kozlowski wrote:
>
> On 20 May 2007, at 18:03, Andrzej Kozlowski wrote:
>
>> *This message was transferred with a trial version of CommuniGate(tm)
>> Pro*
>>
>> 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]=
>>> -6.0968264768052987349716446097396633607810039063851875416961146458846138003732168221294047257351725890381494173`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 =
>>> 2488720838605662428014886339857788161685665826154639846661863271779968897\
>>>
>>> 941302876969944745816129045615885143011927101923791713997993058914014883941331\
>>>
>>> 496588665859636179886756365479484076315048561102041450220571014497428072837453\
>>>
>>> 490447134892293461819188050968748780135755569233537426736962247783202459889540\
>>>
>>> 213301883484666470466149889402655143734621040204402439497074243583844435180857\
>>>
>>> 228403580970629296798899333826598686243987854716724374760335810100582327703252\
>>>
>>> 886711404982379820790899904312876809580414490656116484737937974600066542685289\
>>>
>>> 106532890742345783983687027507936729079442473934078360160815378816949415366223\
>>>
>>> 547953896457883387197030107324924232558604649327195920807344164169408849950012\
>>>
>>> 979654395273385341095562256314722477722302818244400186545582913013684116069229\
>>>
>>> 948450508385560502376379491505913877574694543067098950233734987525958694493166\
>>>
>>> 065786146114295805170616134580156268741967789244572258673205513485511448982113\
>>>
>>> 074128616447024942770432196754923847050903086833932583983456210775092840495926\
>>>
>>> 289398412204946622896060874294857076651762085967637510807753767056601346018771\
>>>
>>> 027068086233850837047631634161338416471812349025685230145549063307448984654469\
>>>
>>> 500345708114334002372857024261410333404070216793731889901563587912181986503488\
>>>
>>> 932240588333472792264516219643268144193209629883467045872736189979709366330108\
>>>
>>> 944683622923025480388609270892579905058376065654372722673382421099596652032752\
>>>
>>> 423655970286505879088423573116299843248723992370681856106228825253081951335763\
>>>
>>> 606805097314767756009989894248180226689216688712554660307978677642033917433524\
>>>
>>> 177703612346235567428057168862868263715487449187865230223959037178478650607885\
>>>
>>> 929852524030200605375426361295649137497579902728693786036767203892699418847034\
>>>
>>> 973900792486513050707875184722293046768355234117849762278847536427384240325375\
>>>
>>> 931710068928003083282083508258941657571106418546338991654633520007125094003937\
>>>
>>> 060577513244349419124583678640310438044741715469307650984762987113625655095113\
>>>
>>> 341410659514797573216487308588020792972361604798011836953448415069774170327604\
>>>
>>> 176428382899037366367969875803830362244613565592323446457417387836546707590791\
>>>
>>> 148857442335097804365308142758237796222541372347526347511124157083242577253654\
>>>
>>> 864546653468558226069365021560451385770280243507694206247762400972408775051143\
>>>
>>> 528825344094380032368218145009068738988932699440006161647412432021399929998924\
>>>
>>> 197063449517037778261055705878691043258271291941546764790768702904202815388755\
>>>
>>> 953467402952252786624210537218217362187375224335225100774863989100606085031055\
>>>
>>> 987180950433574684009505526256479756716140052888061921437953507269705531834507\
>>>
>>> 752244853777872848075149669430514248120843405866305425664958833381695289311873\
>>>
>>> 275612903811562531683996339721232710796969624597692084825522259134899944567445\
>>>
>>> 316144180191492624723899611977533345482296723851296876182987982763612903081830\
>>>
>>> 406428255761789360866674785134042824865250319983289744838888137526494195021927\
>>>
>>> 158720998042457987098509876243983825524393130319382015891243101298654993872084\
>>>
>>> 034846505853704619531981994143584471102830065857739428507878016585984828808526\
>>>
>>> 342887038330953482823346606566055339838200632031259942468414620516606902878898\
>>>
>>> 295905037327168661392320861496592384492793915926275510204303513646878274710211\
>>>
>>> 927798593011178010654392195694992994203684249930039904616401126153259826319808\
>>>
>>> 971152916585811064172283699654029309129460623214205826005262694547534088;
>>>
>>>
>>> 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\
>>> 56658261546398466618632717799688979413028769699447458161290456158851430119271019237917139979930589140148839413314965886658596361\
>>>
>>> 79886756365479484076315048561102041450220571014497428072837453490447134892293461819188050968748780135755569233537426736962247783\
>>>
>>> 20245988954021330188348466647046614988940265514373462104020440243949707424358384443518085722840358097062929679889933382659868624\
>>>
>>> 39878547167243747603358101005823277032528867114049823798207908999043128768095804144906561164847379379746000665426852891065328907\
>>>
>>> 42345783983687027507936729079442473934078360160815378816949415366223547953896457883387197030107324924232558604649327195920807344\
>>>
>>> 16416940884995001297965439527338534109556225631472247772230281824440018654558291301368411606922994845050838556050237637949150591\
>>>
>>> 38775746945430670989502337349875259586944931660657861461142958051706161345801562687419677892445722586732055134855114489821130741\
>>>
>>> 28616447024942770432196754923847050903086833932583983456210775092840495926289398412204946622896060874294857076651762085967637510\
>>>
>>> 80775376705660134601877102706808623385083704763163416133841647181234902568523014554906330744898465446950034570811433400237285702\
>>>
>>> 42614103334040702167937318899015635879121819865034889322405883334727922645162196432681441932096298834670458727361899797093663301\
>>>
>>> 08944683622923025480388609270892579905058376065654372722673382421099596652032752423655970286505879088423573116299843248723992370\
>>>
>>> 68185610622882525308195133576360680509731476775600998989424818022668921668871255466030797867764203391743352417770361234623556742\
>>>
>>> 80571688628682637154874491878652302239590371784786506078859298525240302006053754263612956491374975799027286937860367672038926994\
>>>
>>> 18847034973900792486513050707875184722293046768355234117849762278847536427384240325375931710068928003083282083508258941657571106\
>>>
>>> 41854633899165463352000712509400393706057751324434941912458367864031043804474171546930765098476298711362565509511334141065951479\
>>>
>>> 75732164873085880207929723616047980118369534484150697741703276041764283828990373663679698758038303622446135655923234464574173878\
>>>
>>> 36546707590791148857442335097804365308142758237796222541372347526347511124157083242577253654864546653468558226069365021560451385\
>>>
>>> 77028024350769420624776240097240877505114352882534409438003236821814500906873898893269944000616164741243202139992999892419706344\
>>>
>>> 95170377782610557058786910432582712919415467647907687029042028153887559534674029522527866242105372182173621873752243352251007748\
>>>
>>> 63989100606085031055987180950433574684009505526256479756716140052888061921437953507269705531834507752244853777872848075149669430\
>>>
>>> 51424812084340586630542566495883338169528931187327561290381156253168399633972123271079696962459769208482552225913489994456744531\
>>>
>>> 61441801914926247238996119775333454822967238512968761829879827636129030818304064282557617893608666747851340428248652503199832897\
>>>
>>> 44838888137526494195021927158720998042457987098509876243983825524393130319382015891243101298654993872084034846505853704619531981\
>>>
>>> 99414358447110283006585773942850787801658598482880852634288703833095348282334660656605533983820063203125994246841462051660690287\
>>>
>>> 88982959050373271686613923208614965923844927939159262755102043035136468782747102119277985930111780106543921956949929942036842499\
>>>
>>> 30039904616401126153259826319808971152916585811064172283699654029309129460623214205826005262694547534088
>>>
>>>
>>> Out[401]=
>>> -0.999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 999999999999999999999999999999999999999999999999999999999999999999999999999999\
>>>
>>> 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 cna only guess of course...
>
> Andrzej Kozlowski
>

This is done using RootReduce. o3 is a cubic algebraic number raised to
an integer power, so we stay within a cubic extension of rationals.

In[1]:= o3 = ((2/(27 + 3*Sqrt[69]))^(1/3) + (1/3)*((1/2)*(27 +
3*Sqrt[69]))^(1/3))^27369;

In[2]:= (ro3 = RootReduce[o3])//Short

3
Out[2]//Short= Root[-1 - 294821<<1661>>53132 #1 + <<1>> + #1  & , 1]

Now Element knows that a root of an irreducible cubic cannot be
an integer.

In[3]:= Element[ro3, Integers]
Out[3]= False

Best Regards,