Hi,
while waiting for a cofee, I browsed an old Mathematica book from
1996, and let me assure you, we still have things to catch up on
almost every page. :) But this caught my attention, try it in sympy:
In [1]: e = ((2/(27+3*sqrt(69)))**(S(1)/3) +
((27+3*sqrt(69))/2)**(S(1)/3)/3)**30000
In [2]: e
Out[2]:
30000
⎛ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎞
⎜ ╱ ⎽⎽⎽⎽ ⎟
⎜ ╱ 3⋅╲╱ 69 ⎟
⎜ 3 ⎽⎽⎽ 3 ╱ 27/2 + ──────── ⎟
⎜ ╲╱ 2 ╲╱ 2 ⎟
⎜────────────────── + ──────────────────────⎟
⎜ ⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽⎽ 3 ⎟
⎜3 ╱ ⎽⎽⎽⎽ ⎟
⎝╲╱ 27 + 3⋅╲╱ 69 ⎠
In [3]: e.n(6000)
Out[3]:
504316596065665493215059469242757481077610274680893503167844467712875718226486
210610398076581934068275385782540821330947870035477787176694685987718254576565
515786375180403084108087720197360263045377599299252600787464343711547834066265
665424226316713797047087646002867659572088255372994141399119295425729390551598
734162316493172898778929854821816913956381635362412821599451025319364903651110
221515233814364527904893091337135424027258899192161986823235528295581059873200
107875550497563219302434498746973208148942951387505081508954467615093825332988
410762937088778366046553068685772855386645950482646814439301999016574337736406
592225567911709051797128549077311912778888144108736659703862174018641106917574
995202365941964030217895956126590152078242574779075678185904725211878343979501
048953268614290340265384740790514648773550021687547584525222504190436087068107
827922710445722736236879669324173378416116695919841030467768296491940904387052
300273200377023748044343103233169109426144114438029834875192019905956279051188
804975026500268288352191389685849628968612357880766725385147857934444955391441
537519437285121796122315842862423189949212117527981448636085804805657512086481
790698235946554679103334557030614430688918237659355818871823118397751118724092
116272862912671381160865803740893007151687615872534112762296886660521036261139
972821525679422459991243185253122720544532313520622663794272436418750735995618
191121806899179577732679626160875234377580466842238136144609068158859423168444
990563130856914398161104641435976464505827310532042357277805504151076128744133
166097040034356155594355699310372567007362846011243924380051721179466481515865
177147916094265318673753576167289982789156765819918501461470394696987900055057
821804367427737785530796529257310070440503552865770656588911341336781719293479
609751349965199241672156885820419323697826495675216785846761301665051497508373
554704845376808917106579182046296637409148513289873712928793514345917581372183
082138176837444493669698475749155502722721819149673730689452542949330172497476
348802878923380972880989819941626981662787664453931670154635431610660994478782
022049824681885420029046846763098262298893067982085873006027223547673545004480
573914810092491101933870588744990654481144456437881423922057266818027803279734
701366817533469290959062710889715842108254898313134138767612788549048657824912
624883639420318879265232485613570477399709046710448077813242383525775842418033
574517057365270541979506919519088585109853839062549850746730941961116578083869
871331034408636054845107081922257029648252724270305699507491876725741414282227
189033471727764395434254185950990747290625952762962184681171936790178602417989
512006492961330651781076337449499729308205550243207661846458602761611981567677
373636018999149195717505734613349483793082747826653826109388871646107637707669
952322121063841012111950272802819214732121339193913851711740180545436004534753
223529995802192329202767788406529942228608625210506805680779347684445472832022
947989505073864225065042453778536680082426933681881662317694955672636170867065
177268568641882560696136399378938092363061534206649910677121667830726528065065
115430787822086187067816894109743170406015960678013961777060204838325510730464
182561678677866776664114588829507728845970964346546857260085067375205588321834
061968066477136954754650582659722182787527038013350885360773847158970095196369
406644986845955931896822762457057294441289666846719625035790435931009430361335
058057283719525424750533492243401220689154316788170873900888499149058827913507
680455068280339445423082211488881873764963543994136415115212683005334092753230
4714488156842955217262531239581967894186558682010795291845929630207675781253.0
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
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000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000
000000000000000000000000000000000000130894811446517479512360354894549290725102
417205882279175127063157759595730280694948193177497539095028808225225138522519
220774088509824569077156937068499526850929967148021404867831962245917243427093
569827360443678604397329589365618912756472236130577656466210931482151622206079
461642426944286257851344424055950819293859084089972930323131973513431224561849
298256944105389850990056499442812694858850038546319831776522919282478332247247
3947007368525562887408463240932824486779007274030664120841078472914502641
Look at the number of zeros, very nice number. Btw, the numbers are
correct, or at least the same as in the Mathematica book. However, try
this:
In [4]: e.n(3667)
Out[4]:
504316596065665493215059469242757481077610274680893503167844467712875718226486
210610398076581934068275385782540821330947870035477787176694685987718254576565
515786375180403084108087720197360263045377599299252600787464343711547834066265
665424226316713797047087646002867659572088255372994141399119295425729390551598
734162316493172898778929854821816913956381635362412821599451025319364903651110
221515233814364527904893091337135424027258899192161986823235528295581059873200
107875550497563219302434498746973208148942951387505081508954467615093825332988
410762937088778366046553068685772855386645950482646814439301999016574337736406
592225567911709051797128549077311912778888144108736659703862174018641106917574
995202365941964030217895956126590152078242574779075678185904725211878343979501
048953268614290340265384740790514648773550021687547584525222504190436087068107
827922710445722736236879669324173378416116695919841030467768296491940904387052
300273200377023748044343103233169109426144114438029834875192019905956279051188
804975026500268288352191389685849628968612357880766725385147857934444955391441
537519437285121796122315842862423189949212117527981448636085804805657512086481
790698235946554679103334557030614430688918237659355818871823118397751118724092
116272862912671381160865803740893007151687615872534112762296886660521036261139
972821525679422459991243185253122720544532313520622663794272436418750735995618
191121806899179577732679626160875234377580466842238136144609068158859423168444
990563130856914398161104641435976464505827310532042357277805504151076128744133
166097040034356155594355699310372567007362846011243924380051721179466481515865
177147916094265318673753576167289982789156765819918501461470394696987900055057
821804367427737785530796529257310070440503552865770656588911341336781719293479
609751349965199241672156885820419323697826495675216785846761301665051497508373
554704845376808917106579182046296637409148513289873712928793514345917581372183
082138176837444493669698475749155502722721819149673730689452542949330172497476
348802878923380972880989819941626981662787664453931670154635431610660994478782
022049824681885420029046846763098262298893067982085873006027223547673545004480
573914810092491101933870588744990654481144456437881423922057266818027803279734
701366817533469290959062710889715842108254898313134138767612788549048657824912
624883639420318879265232485613570477399709046710448077813242383525775842418033
574517057365270541979506919519088585109853839062549850746730941961116578083869
871331034408636054845107081922257029648252724270305699507491876725741414282227
189033471727764395434254185950990747290625952762962184681171936790178602417989
512006492961330651781076337449499729308205550243207661846458602761611981567677
373636018999149195717505734613349483793082747826653826109388871646107637707669
952322121063841012111950272802819214732121339193913851711740180545436004534753
223529995802192329202767788406529942228608625210506805680779347684445472832022
947989505073864225065042453778536680082426933681881662317694955672636170867065
177268568641882560696136399378938092363061534206649910677121667830726528065065
115430787822086187067816894109743170406015960678013961777060204838325510730464
182561678677866776664114588829507728845970964346546857260085067375205588321834
061968066477136954754650582659722182787527038013350885360773847158970095196369
406644986845955931896822762457057294441289666846719625035790435931009430361335
058057283719525424750533492243401220689154316788170873900888499149058827913507
680455068280339445423082211488881873764963543994136415115212683005334092753230
4714488156842955217262531239581967894186558682010795291845929630207675781252.9
99
as you can see, the last 4 digits are wrong. Or not? If you try:
In [6]: e.n(3668)
[...]
4714488156842955217262531239581967894186558682010795291845929630207675781253.0
000
so I am confused about those digits at the end. But apparently they
can't be trusted.
Ondrej
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