John Nash and the Beautiful Buddhist Number (50,000 digits)
by Jason Earls, author of Red Zen and Cocoon of Terror http://becomeguitaristfromhell.blogspot.com/ http://www.youtube.com/user/zevi35711
John Forbes Nash Jr. was born in 1928 and is now one of the most recognized mathematicians in America still living today. In 1994, Nash won the Nobel Prize in Economics, and a best-selling book, A Beautiful Mind by Sylvia Nasar, was written about him and later made into a Hollywood movie (which turned out to be a quite mediocre and inaccurate film) by Ron Howard. Nash’s mathematical work involves the subjects of game theory, algebraic geometry, partial differential equations, gravity, and he also possesses a recreational interest in computer programming and number theory. He was born in West Virginia, the son of an electrical engineer and an English teacher, and in his youth Nash was quite precocious, reading encyclopedias and other books while performing sophisticated science experiments in his room, always preferring to work in solitude. At a young age Nash also read E.T. Bell’s famous book, Men of Mathematics, and was particularly taken with the chapter on Pierre de Fermat – the French lawyer and number theorist – which helped to instigate Nash’s interest in mathematics. Young John attended public schools but was also tutored by his mother outside of the classroom, which later helped him win a scholarship to attend the Carnegie Institute of Technology where he studied chemistry and chemical engineering. Later he switched to mathematics and headed off to prestigious Princeton University where he wrote a doctoral dissertation on non-cooperative games, which introduced and elucidated his now famous and widely-referenced ‘Nash equilibrium.’ He also made valuable discoveries in the area of algebraic geometry, writing a paper titled, “The Imbedding Problem For Riemannian Manifolds,” which solved the longstanding problem of the same name. Nash then taught advanced mathematics courses as an instructor at M.I.T. for a couple of years before developing symptoms of schizophrenia. He was admitted to a hospital where he stayed for a few months and psychiatrists diagnosed him with “chronic schizophrenia.” He suffered with this mental disease for roughly 25 years, yet still managed to do a little mathematics while being ill. Nash said in an interview, “Though I had success in my research both when I was mad and when I was not, eventually I felt that my work would be better respected if I thought and acted like a ‘normal’ person.” While sick with schizophrenia, Nash regularly attended the Princeton University campus and would wander around the library, whistling, reading, and also writing nonsensical equations and other phrases on various blackboards, where he became known as “the Phantom of Fine Hall.” Concerning his whistling, Nash was an excellent whistler throughout his life and could whistle extremely long passages from Bach’s works along with other famous classical pieces. Nash also developed the urge to travel widely while suffering from schizophrenia and soon found himself heading off to Europe. While there he sent a variety of postcards back to friends and family as he traveled around hoping to legally give up his U.S. citizenship. On one of his postcards, Nash mentioned a number that he dubbed, ‘The Beautiful Buddhist number,’ and for the remainder of this article, I would like to discuss this somewhat curious mathematical entity with the interesting name.
The Beautiful Buddhist number is
22 * Pi + 4 * e
which has a numerical value of
79.9881656928116321876193043175597134433667... From the name Nash gave this number, it sounds as if it could possess supernatural or mystical properties. But we also must keep in mind that, unfortunately, he was suffering from a serious mental disorder when he discovered it (no disrespect intended by this statement). Nevertheless, the Beautiful Buddhist number seems interesting, so let’s try to examine a few of its properties. On a very basic surface-level, we can see that the Beautiful Buddhist number includes Pi and e, two of the most important transcendental constants in mathematics. Also the numbers 22 and 4 make an appearance. Why did Nash choose 22 and 4? Recall that 4 is the second square number (the first nontrivial square, in my opinion), while 22 is a semiprime (it has only two prime divisors); but in numerology – a field Nash was keenly interested in while suffering from schizophrenia – 22 is known as the “Master Builder” number and also the “Spiritual Master in Form.” Quoting from Wikipedia, “This ‘master number’ includes all the attributes of the number 2, twice over, and also those of the 4. People who are 22s are said to find themselves feeling as if they live in two worlds, one that is overwhelmed by the mundane, and the other by the fantastic.” Could this numerological answer be the reason Nash chose 22 and 4 to multiply and add with Pi and e to arrive at his Beautiful Buddhist number? Perhaps. Only John Nash would know for sure. Nevertheless, concerning the Beautiful Buddhist number, 22 * Pi + 4 * e = 79.9881656928116321876193043... I have noticed a few other properties in its decimal expansion. After writing a computer program to analyze the number, I found that the first 1 occurs in the fourth position after the decimal point; and the first occurrence of two 2s or 22 occurs at the 85th position after the decimal; while the first occurrence of three 3s or 333 is at the 122nd position. I have never found the occurrence of four 4s or 4444 in the decimal expansion of the Beautiful Buddhist Number. (Perhaps you would like to try?) So my questions to the reader are: Where do you think five 5s will occur? Will anyone ever find a large quantity, such as, say, nine hundred and ninety nine 999s in a row somewhere in the decimal expansion of the Beautiful Buddhist Number? (I realize these questions are not especially serious mathematics – that instead they fall more into the realm of ‘campy mathematics’, but that is all right with me since I love campy things in life and in art.) But what can we make of the Beautiful Buddhist number in terms of real mathematics? Should we consider it a significant number worthy of further study? Or is it something not worth investigating since John Nash was probably in the grip of full-blown schizophrenia while he concocted it? – (Again, no disrespect is intended here.) Although we actually need a real mathematician to answer these questions, I will state that I think the Beautiful Buddhist number
is worthy of more study and I’m quite intrigued by its formulation and would like to know more about it, so I hope a few real mathematicians will study it soon. One more thing. What exactly is “Buddhist” about the number? Although there is really no way to tell, Allan Watts, the popular Buddhist writer once said, “Just as the highest and the lowest notes are equally inaudible, so perhaps, is the greatest sense and the greatest nonsense equally unintelligible.” A few years ago I made a sequence from some of the data available on John Nash’s web page at Princeton University. The description of my sequence runs thus: “Even integers that can be expressed as a sum of two primes such that the smaller prime cubed is greater than the larger prime.” You can find this sequence by typing ‘A093161’ into the Online Encyclopedia of Integer Sequences database, if you are sufficiently interested. Now I will end the main portion of this article with a quote from John Forbes Nash Jr. himself: “So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists. However this is not entirely a matter of joy as if someone returned from physical disability to good physical health. One aspect of this is that rationality of thought imposes a limit on a person’s concept of his relation to the cosmos.” (For more information on Nash’s Beautiful Buddhist number, see my novel Red Zen.) Below you will find a listing of the first 50,000 digits of the Beautiful Buddhist number. This is the most ever computed, as far as I know. There are many reasons I am providing such a large quantity of digits in this article, with the primary reason being that perhaps there are patterns apparent in the pages that follow, and someone with more intelligence than myself will be able to spot one or more of them. If someone does spot a pattern, or anything interesting about the digits that follow (I noticed 5 zeros in a row make an appearance on page 8 of this article), perhaps they can tell us why John Nash thought this number was interesting enough to mention on a postcard written to (I assume) one of his mathematical colleagues. But even if nothing is discovered in the decimal expansion of the Beautiful Buddhist number provided below, I still believe simply flipping through the pages and examining the digits in a casual manner – observing them, contemplating them, perhaps even meditating on them in a Buddhist way, attempting to become one with their numerical nature – can be quite rewarding in and of itself. Enjoy.
Fifty Thousand Digits of 22 * Pi + 4 * e
(Computed by Jason Earls)
79.988165692811632187619304317559713443366715161052166361316651541668267459710 788348102294871627241205317123471020222985923334235796628983389559244159868967 106858779573954421427375247501227889789472479263729178216373344431183064000679 979266615012284673533327672990037316844964568030443089516036461076300147725192 693367985172931792264190423325323979206520585440723086586508583825251659374020 851550495974298607224148771457597883183027412076597148540422844721314159060362 269199306183899333562196946658053367285526835394443066197876617653774524992867 580687056239054937723201162206847870418019287738764685373217451585469462899799 366320257954104531325877149053552909142079014741047352085189986875481442941321
909166560839005244545404311276899128950511923176822928472649268347187508514217 365250891899407971988964488348952131290826049983617582227031716979965907158805 325406179516079487915596999638312672276068746235162486945182682943764579250624 212161104078073241334162524454584188235630275370514928758589384518246589289822 718026276407731079396911736241838775843756704180936828699528295479622135547045 555441248093433420756274180218297114752890488619978096432663059406146597159694 489058931879596188636175462513921902322706227089467784272450935994007722787346 113953641931084813402430834534329177532600818691383105933955915815554698296612 067222366566307969552627738167718572516474351371433312462188807599260518294686 472031614895416632403563459123511907701089204223948756157642005571174302579490 992039221161567010211052802739345942650708637261632922504503394832850975519445 549271828664140460331846585844263668423669733057641968650663733618900334720729 565834766312771085400959514797065465723308289192450253561502086464737908883115 203644266917954470593233649087124437503343745475341783716520720159105060334109 570313684453631953440208339797537888960633629165413303803597226063269976267572 692073927370461485065759472084951193133786648293481921343781259881360163595093 056899606481569674223397111631887225809634857662907770642836687521097282264905 844114480242423422652566578086474957032846019567151133061067555211823486226153 542094033905240044898111386033170863640192809131645137741144972138535392125719 122397431804195106192895603013318424571968103985782394906967779184017356520961 670744169812269076917898874498206411921777907381162895037163288489373925109318 057656373989522592642404665685071251551790145862486277446059328238165448427614 181334046167994747535168200976770592457107881948980577259801513494889776564384 024950489134570700547491462816257189778298768540822751592790928476463398403166 712392715178879032824265032410766801329462522308576932161775613100621268560255 599001685489307670696763051923092441163699435553527611053696616190542986891950 457925900294190286831656455832087127004071561226015535449117569138227279987765 545816439370754704786272098864457644950814461462150621504125527376803783583965 954069609022877205226210642365008912601262502021883277548209959998529684307986 518793731211486909276142612736166014638245395482152940509561966339780303935362 487144991793146195431066006177686174931905491780719415926741004393726260140303 209997043938755439114340572040560123110484625205634955690452770348801630292958 374340151829977670992361939420999873221631263309863826573464217335176969149905 066002173867712452881190520591511125152836628653559734146352358901690436091191 408077541089682905895305624135507131940932885210190362042686486236431873593906 151869001376739464520154782654475651375208470175259166634349768281375851235429 598479432635236819753131057591119354039812219260896642600348366475956537522012 740196738990264605289538675286293950209275422747452656167697830949468273730450 119198269002342940243180555047489681364468577460895917609701995727438307640066 055722989528422586511792978331575479285655386252614464508290510679258172326146 391451721353910930599990691275406845229218290837529259440520005836737471295613 206874965090412412466829434463634917504693940824956417234342072825305604544357 852958362132657670908871552942325450480385109918135670341138059859211108927594 921421185113885716410283340811976483563908323948053733348152446318511582009582 710922068386376907811653908425443233711212960294651642778328594127666294728462 951380103947709835010820227350891298465721950211718933787163778202374777300456 653475334391125669245578958098075294935848787542840587143074723314279513956752 723741960745307526950576432549426351842603636350124875874140279514486157138111 008699109410460679276746084626241196151312925046992371508188834956742846629112 155885046075303057768314245091301915256294245146660317759879488591447616843603 633987676069881946483803803028212080658440290204038857413639012163853324330289
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(Thanks for reading. If you now of any magazines that would like to publish this article, please contact the author: zevi_35711@yahoo.com. Also, you would be helping out the author greatly if you purchased one of his books from Amazon.com or another online book store of your choice. Thanks again.) http://becomeguitaristfromhell.blogspot.com/ http://zombiesofthereddescent.blogspot.com/ http://www.youtube.com/user/zevi35711 Bio: Jason Earls is the author of Cocoon of Terror (Afterbirth Books), Heartless Bastard In Ecstasy, How to Become a Guitar Player from Hell, Zombies of the Red Descent, If(Sid_Vicious == TRUE && Alan_Turing == TRUE) {ERROR_Cyberpunk(); }, Red Zen, and 0.136101521283655... all available at Amazon.com and other online book stores. His fiction and mathematical work have been published in Red Scream, Yankee Pot Roast, M-Brane SF, Scientia Magna, three of Clifford Pickover’s books, Mathworld.com, AlienSkin, Recreational and Educational Computing, Escaping Elsewhere, Neometropolis, Thirteen, Dogmatika, Prime Curios, the Online Encyclopedia of Integer Sequences, OG’s Speculative Fiction, Nocturnal Ooze, Bust Down the Door and Eat All the Chickens and other publications. He currently resides in Oklahoma with his wife, Christine.