Chapter 15

1.002008392826082214417852769232412060485605851394888756548596615 9097850533902583989503930691271695861574086047658470602614253739 7072243015306913249876425109092948687676545396979415407826022964 1544836250668629056707364521601531424421326337598815558052591454 0848901539527747456133451028740613274660692763390016294270864220 1123162209241265753326205462293215454665179945038662778223564776 1660330281492364570399301119383985017167926002064923069795850945 8457966548540026945118759481561430375776154443343398399851419383

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 [email protected]

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: [email protected] 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: [email protected]

The request was sent by Joe Keane on Sun Sep 10 05:02:26 PDT 1995 The email address is: [email protected]

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: [email protected]

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: [email protected]

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: [email protected]

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: [email protected]

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 [email protected]

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

The Hard hexagons Entropy Constant

The hard-hexagons entropy constant is algebraic (see below z number).

The value is :

1.3954859724793027352295006635668880689541037281446611908174721561357608803586 977746898378730852754279026689685607685657184842212457119511639349818266947083 252547173794947534862281229126187281554340126162747356973585709823756812898414 948800016934903723995652094568253572538633572005211925074739811015138086289661 268136787831885630404682747107477204686894756657580905530270066675404962427719 060854536142216836296933016900330937276956621269398726823104923047442882514781 702966107270054292812280795061336321550953581179745072336957434963259935073449 490894249329307540816210555328068610619705545037955077580725537613858033619505 210958967729699416630942601615566925218549336476968551824281894615092855649748 501359906929152571833851080212811049755339847366927914398892041851355831303575 673710465224807454744982583885183287167357146092090743402851746571565499082292 999884612996137479952358336507860770516087879631202738350102895965881076822440 14681214726789035888008851819053742866660552775722734105313225337

Taken from The Favorite mathematical constants of Steven Finch, Mathsoft Inc.

The constant is given by this (see z below)...

124 1/3 a := - --- 11 363

2501 1/2 b := ----- 33 11979

/ 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\1/3 c := |1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| || \ 242 \\11979 / \11979 / //

1/4 7/12 3 11 z1 := 3/44 ------------------------------------------------------------------ / 31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\\2/3 |1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| || \ 242 \\11979 / \11979 / //

1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 z2 := (1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )

31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / /

1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 z3 := (- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )

31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / /

1/3 1/2 z4 := 1/(1/33 (1089 + 372 11 )

/ 124 1/3 / 124 1/3 15376 2/3\1/2\1/2 + |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2 \ 363 \ 363 131769 / /

1/4 7/12 z := 3/44 3 11

1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 (1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) )

1/3 1/2 1/3 1/3 2/3 1/2 1/2 2 / (- 1 - (1 - %1 ) + (2 + %1 + 2 (1 + %1 + %1 ) ) ) / ( /

2/3 1/3 1/2 %1 (1/33 (1089 + 372 11 )

/ 124 1/3 / 124 1/3 15376 2/3\1/2\1/2 + |2 - --- 11 + 2 |1 - --- 11 + ------ 11 | | )^1/2) \ 363 \ 363 131769 / /

31 1/3 // 2501 1/2 \1/3 / 2501 1/2 \1/3\ %1 := 1/4 - --- 11 ||----- 33 + 1| - |----- 33 - 1| | 242 \\11979 / \11979 / /

evalf(z);

1.395485972479302735229500663566888068954103728144661190817472165

End of Project Gutenberg's Miscellaneous Mathematical Constants, by Various