.8116869215447793 a href="/projects/ISC/data/flajolet.txt">a Flajolet constant .8711570464148937 a href="/projects/ISC/data/flajolet.txt">a Flajolet constant 1.628160129718917 a href="/projects/ISC/data/flajolet.txt">a Flajolet constant 2.302566137163630 a href="/projects/ISC/data/flajolet.txt">a Flajolet constant This number, the Product[Cos[Pi/n], {n,3,infinity}] is the limit of an interesting figure in geometry.: If we take a circle, inscribe a triangle, then incribe another circle inside the triangle, then inscribe a square inside the inner circle, then inscribe another circle inside the square, then inscribe a pentagon... The radius of this figure (the number of sides of the polygon increase with every step:triangle 3, square 4, pentagon 5, ...) approaches a limit: Product[Cos[Pi/n], {n,3,infinity}] Is there any way to get an analytic solution to this? Like this would be the square root of Pi or some combination of radicals and irrational numbers? Anyway, Thanks, Mounitra Chatterji mounitra@seas.ucla.edu mentioned in december 1995. By Mounitra Chatterji .1149420448532962007010401574695987428307953372008635168440233965; maple routine --> product(cos(Pi/n),n=3..infinity);evalf(",64); ------------------------------------------------------------------ The request was sent by achim flammenkamp on Tue Feb 27 09:05:13 PST 1996 The email address is: achim@mathematik.uni-jena.de The number is 1.60140224354988761393325 (to 24 digits of precision). -int(sqrt(x)/log(1-x),x=0..1); ------------------------------------------------------------------- .283265121310307732587685540450858868452123075913479495609303244760289207466703551200728343246718266 1721794706326872389237418265273196389116929121819750888062495294277256191719424273967384545908106616 5124702322513598413388920213387535350692362866707758376138858482266928332718882186473891252470626193 1134162075403008037881499615240658150936661712754874529120769279078826146925069339158824377250780006 81691683658433538480533518043146405030754456294577975558177142447872562829157 There is a pattern in the binary expansion of this number. The request was sent by B.J. Mares on Sun Dec 3 15:20:18 PST 1995 The email address is: bjmares@teleport.com ------------------------------------------------------- The request was sent by Joe Keane on Sun Sep 10 05:02:26 PDT 1995 The email address is: jgk@netcom.com The number to be tested is: 1.38432969165678691636600070469187275993602894672280031682863878069088210808356345 The number of correct digits in the number: 79 The hints given by the user: It's log((3+sqrt(7))/sqrt(2)) or 1/2*arccosh(8). -------------------------------------------------------- The request was sent by (Mr.) B.J. Mares on Sat Dec 9 19:10:27 PST 1995 The email address is: bjmares@teleport.com The number to be tested is: .86224012586805457155779028324939457856576474276829909451607121455730674059051645804203844143861813\$ 451257229030330958513908111490904372705631904836799517334609935566864203581911199877725969528883243\$ Another binary pattern. --------------------------------------------------------- The request was sent by Jon Borwein on Sun Nov 5 06:09:28 GMT 1995 The email address is: jborwein@cecm.sfu.ca The number to be tested is: .01118680003287710787004681 The number of correct digits in the number: 20 The test(s) to be performed on the number: algebraic -------------------------------------------------------- 1.456791031046907 The number of correct digits in the number: 16 The test(s) to be performed on the number: algebraic gamma_multiplicative gamma_additve zeta_multiplicative zeta_additive psi_digamma linear_dependence_salvage The hints given by the user: p(0)=1 q(0)=2 p(i+1)=sqrt(p(i)*q(i)) i = 0,1,2,.. q(i+1)=(p(i) + q(i))/2 i = 0,1,2,.. x = lim p(i) = lim q(i) i->+inf i->+inf -------------------------------------------------------- The request was sent by Olivier Gerard on Mon Jan 29 18:48:42 PST 1996 The email address is: quadrature@onco.techlink.fr The number to be tested is: 1.062550805496255938 This number arises in the study of generalized Zeta functions on non associative sets. -------------------------------------------------------- The request was sent by Michael Mossinghoff on Fri Feb 9 14:40:28 PST 1996 The email address is: mjm@math.appstate.edu The number to be tested is: 1.296210659593309 (see below for 2500 digits of it). As I mentioned in the original note, it would be interesting to see if this number satisfies a simple polynomial of degree > 34. The simplest polynomial I know of that it satisfies is x^38-x^36-x^34-x^29+x^28-x^24-x^14+x^10-x^9-x^4-x^2+1 I found this during a search for polynomials with height 1, degree 38, and Mahler measure < 1.3. I also have a second new Salem number that would be interesting to try. Thanks for running this! Best regards, Mike Mossinghoff mjm@math.appstate.edu 1.2962106595933092168517831791253754042307237363926176836463419715400357507663\ 555372700460810162259842255138960885075885472138523375229647035948031308222869\ 213377761985420998401465270339786283142588526265385851765349326219909024384324\ 298668143261669279113959085262729367911041451897621484638159134108808507417558\ 371227480609429111967509190900525542468572422201267290352457473788303514632978\ 531591219560940258062757424400763572149784569551257493407108061275808255266204\ 988526404732083078237046586577078037338486088388181584983281574252897177808263\ 147692481736785688370028996889741999268557158363474402864561998038209817582814\ 010732290535268946721928114002527443568020359790313185377702725896115435126307\ 841519785171242185997657977732689357703555840184684554577244752237497568339160\ 938205575175811976414747122955198011255949965359970687280700475477368518212756\ 924749820065045209604606889253335548989681523027453599219856774850675170030081\ 340461412329460883636590018878175768282781839837697211776636498168350816554156\ 904601023147786817236407289883278093415918634119620218433047846657184261144649\ 040715513536648841284787099601551612909626813632800691067564404454541790010887\ 679088108728482285977923782153457884089162309486388513634809308430291906873755\ 353865787785568433558148544650806363798445573460997103012214477139122206697676\ 151378710063572151250547043624062114013563819037462333697027524356258777528864\ 271328965733484293667236211401267719087175146826163754038706366216877272628132\ 296182344392845125506127123945469182368918766036231606918375224969603018840277\ 778903237698826183111400261578682603995590568903906955569848314084496482503972\ 906016618276547328327517227379822958377122743985938689837061722495995392321936\ 345285971817821600170724492417762482659737742843585759061520292400466743607983\ 593438732628413114256276767063139352552076489085606199932942061150333621663624\ 667294211959583161911171198313494502505440901133068426838051637173543721800267\ 607050254597479936347302850855318828765200608121163125879643065717811879123723\ 939826702878343201235748915166745912187493987556824139288848294746007488299743\ 663817162198495190194616103659925459932420514340386336983265209362290719538034\ 616103846861918706369114431911997889483119661422295458652413962075819025018423\ 406629086461013112957825351840936858715307617702746177132615020866202346765384\ 189199689332174745118809280247719860161398327812075021357273956644275172873038\ 687900608249173662145494837168975704911668609774430992557238265593517876057742\ 2513 ------------------------------------------------------------- Reference Philippe Flajolet and Andrew Odlyzko in Random Mapping Statistics you can have the article at ftp://netlib.att.com/netlib/att/math/odlyzko/index.html 1 - ln(1 - 1/E) > evalf(",1024); 1.4586751453870818910216436450673297018769779066921941448349981657928142090774\ 201612200442809516952542077265289812147224950456505217508488257192318776903978\ 283958471454981649855439295026537053597338520354935148025543820985296873219986\ 302608076828991375664708977028227357407155020168390466081440332929613402809962\ 987761600422067245386552208829277426092542078462258992350164685882837621214882\ 780180315165656808973787662538495808236640442271087689278355793100958663124347\ 608912549488795731777070799343730722066801620056545869945636645492898791927486\ 575158188313946857834776772734408679626984363705284330037652725380287794676349\ 373789251316549424606319247455867160631085208147788915528328222030175460874293\ 072958579419651653681072447431245769874928136318703222181432813223236987618651\ 560035148342838332185451812183617068075562954967559891795834498316055598164437\ 208325189384466039982301475617199617179127040273935951240040637361969048372804\ 683416371677229307327020903657448359390542480371335759362920019292630614667717\ 96831954446 1 + exp(-1) ----------- exp(-1) - 1 2.163953413738652848770004010218023117093738602150792272533574120