1000th digit!

Just finished posting this and realized there is a cute alternative way to see that $y < 10^{-1000}$ As I pointed out, $x$ and $y$ are multiplicative inverses. So it suffices to prove that $x > 10^{1000}$ But we know that $x \approx x + y = 2 \sum_{i=0}^{1500} {3000 \choose 2i} 2^i$ If we pair the first and last terms, second and second-last, and so on, we find that the middle pair has the smallest average, namely 2^{750}, so $x + y > 2 \sum_{i=0}^{1500} {3000 \choose 2i} 2^{750}.$ Finally, since the even binomial coefficients have the same sum as the odd ones, namely \frac{1}{2} 2^[3000} we have $= 2^{3750}$ and it is easy to see $2^{3.75} > 10$ . Using a calculator, I find $(1 + \sqrt{2})^3 \approx 2^{3.8146}$ so this wasn't such a bad approximation.

Funny that you can't add a solution if you didn't get the right answer...

Consider this recurrence relation:

$a_{n} = 2 \cdot a_{n - 1} + a_{n - 2}\\ a_0 = 2\\ a_1 = 2$

The solution is: $a_n = (1 + \sqrt{2})^n + (1 - \sqrt{2})^n$ .

(I didn't know about Pisot–Vijayaraghavan numbers, but the characteristic polynomial of the recurrence relation above and the minimal polynomial of $1 + \sqrt{2}$ are the same and that's the link between Pisot–Vijayaraghavan numbers and recurrence relations).

We are interested in $a_{3000}$ . By definition, this is an integer, and it equals $a_n = (1 + \sqrt{2})^{3000} + (1 - \sqrt{2})^{3000}$ .

To estimate the second term: $(1 - \sqrt{2})^{3000} = (\sqrt{2} - 1)^{3000} = 10^{log_{10}((\sqrt{2} - 1)^{3000})} = 10^{3000 \cdot log_{10}(\sqrt{2} - 1)} \approx 10^{-1148.327056013589} \approx 10^{-1149} \cdot 10^{0.672943986411} \approx 4.7091658552 \cdot 10^{-1149}$ .

With the recurrence relation, we can calculate with a computer that:

$a_{3000} = 212351832735661211850036721375886776541419741846830798047797649326633830265800348754922726611482329696978386941712386628313233722329396831729262976271935913028968686217854171694479750950007138598702047874928937006048825941728804726445558472950595484372408258805255973963893612565744659452588885634672251124694961267804898152795815102383284947073021158623268979166708897371716288813328574125896002105522342531264500667666010593004805641201407485523369661123270309321721319990425609586215891216379523104027105154328230715943028688427964504737016740719705648004760033477439060120822056401856842664843498264757732196936579380206418720952006865612562210412207828131451616365295898234402537336762320382508754921202825754807404557820007055121102645156842126064526301550033503710326243580686202343689290385612858013208292479048892324242891437926936379579771661271219677064396120739803309230502981355688484907132471136023854101013706580243115362738647939748123348684831357244274136458378908357255958491925980356588743876497779219636407873285945202016359307298985412901504489975280743182186498490579824057349140600741150807126892554341838245583515532450000002$ .

So:

$(1 + \sqrt{2})^{3000} \approx a_{3000} - 4.7091658552 \cdot 10^{1149} = 212351832735661211850036721375886776541419741846830798047797649326633830265800348754922726611482329696978386941712386628313233722329396831729262976271935913028968686217854171694479750950007138598702047874928937006048825941728804726445558472950595484372408258805255973963893612565744659452588885634672251124694961267804898152795815102383284947073021158623268979166708897371716288813328574125896002105522342531264500667666010593004805641201407485523369661123270309321721319990425609586215891216379523104027105154328230715943028688427964504737016740719705648004760033477439060120822056401856842664843498264757732196936579380206418720952006865612562210412207828131451616365295898234402537336762320382508754921202825754807404557820007055121102645156842126064526301550033503710326243580686202343689290385612858013208292479048892324242891437926936379579771661271219677064396120739803309230502981355688484907132471136023854101013706580243115362738647939748123348684831357244274136458378908357255958491925980356588743876497779219636407873285945202016359307298985412901504489975280743182186498490579824057349140600741150807126892554341838245583515532450000001.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999952908341448...$ .

To get more digits, we can remark that

$(1 + \sqrt{2}) (1 - \sqrt{2}) = -1 \Rightarrow (1 + \sqrt{2})^{3000} (1 - \sqrt{2})^{3000} = 1 \Rightarrow (1 - \sqrt{2})^{3000} = 1 / (1 + \sqrt{2})^{3000} \approx 1 / a_{3000}$ .

The fractional part is then $1 - 1 / a_{3000}$ (the 1 is borrowed from $a_{3000}$ ). If we want to use only arbitrary precision integer arithmetic to calculate the digits, instead calculate $10^k - 10^k / a_{3000}$ . Because we know the first digit after the decimal point is not 0 (it is 9), the number we calculated can be put right after the decimal point. The last digit we find is likely wrong though, so we leave it off. These are the first 10000 digits after the decimal point:

$0.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999529083414483728428493999500285716007029768246046223642942779547405733289828412546046116083818053025319898896313108009211459549222959923017560073605540854889556967317112269549068529519118370484685361445960507491910511846886973363595835012870632906762166451663511575432666263881168765857197313311970522318687540230641194260069360342981495745513776648796498718219012601496816456849757593273683146512236212545005379512089651007924314664099797821398821828502470590178444509406550359794817813507202111624654770556800613338455029215890671255934974207964241668195018445924662959965562196296733844961738381678486925425432072276950682100032197840228869260457696099026749913449827081047414371433263868298395956797455261725469537284505965194000613090031463915588137011662238038545552542741742289932187144718929051552387773317585937648904018057478244442545420225067821143009893722981349269349772293801624716458453855590197850413047406687757354625817159604029534933318928788931859539395004939997145908979182142820171165731099432773119909226154118727255062263330878047492360180115979656577685257478652471881232548615894325830050825905845912139198869843417674774837624930422321898985537376509160957421126604927664070336742870155353324237679912424446894380798453627446081939396573705238251775230980265907915440665237415698508501472190964900131306790999231540924165429482683105513160108222461543205261973030162177098216856247176352207397958239241231871821569326611821981884586363267582345479813976863520782178550173709925567813070649096841471684371347769899240091996794517510494639020713250787348309132980929193004789220399150499814367750697646312945552444277736567759760338741516518254771729773553652721408167446127765553410873994802854118160395829553403926224281865443290597910447527031883054220934501632317556124992148351279412033742996093770822853198159678767866494844397531010696720313974285990207251175069521095868117538773654737384374136819418903777602684243247617559503949431837587886680287353770920344915528708811198251818371851029661840090187022888827075645636245069341796887509748575119261012438076414784113290855875686217745270682730612645549664116863089466622478743244436653433178481826483083048436331930924577411087591473407118984471110362846887050801564628248514980961679264193875035571444331321738739659593543975659653824580767605569834138382015260084348320623192765389388937268423377891169843559382241915516974946750678981083466934169858590172472394032105568169224885940948548552354139329617121004336075447446186597498395913761002689452694821938205599325779252169391022149675749127605201162475355821132691860941920900099920272937128332501623226982756394754089375284607260683306112503818303110546081306709578376193241895739239414631290663827424001938553196372004355034983003038630828148947950811053122837817672848898056461092165842625307547729182032570748984932199914639645344966986268121426652891814673875270845044785062215722949047569779219500792212443310372553569559947523562106330104477510704241418925124322531505336183126466879477342000940289044805531552065588816518506710468707726951955111292944303982981769860172572310044313603150168684014573308261775629016615960390510083060691497415840721385101844526374838756357252019597390925591323557801625112825471830243820798453723746268779804810298632613182456254619671965259503520803819985806759727125571492128919159768979302985657188403831510718818177535847368890632883466247004691044905968670630667514157081783932686023900058754487274394957012224229228659599535087763188285184017595724678629449923829128103640027438830049032894732983492251035022306306084413935823021137499731377106568111327710984919178111737133024296612620673531812949282681632204795174689514647469254490731408864995900484239437777435147161807344858249525286804003656378711703067521563237948759554321326314290283150514965860395108241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605$ .

It would have been an interesting question to ask for the 1150th digit to the right of the decimal point, which can still be done without a computer.