John Forbes Nash and the Beautiful Buddhist Number

John Forbes Nash Jr. was born in 1928 and is now probably the most recognized mathematician in America still living today. Nash won the Nobel Prize in Economics in 1994, and a best-selling book, A Beautiful Mind, by Sylvia Nasar was written about him and later made into a Hollywood movie (a quite mediocre and inaccurate film) by Ron Howard.

Nash's mathematical work involves the subjects of game theory, algebraic geometry, partial differential equations, and also 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. In his youth, Nash was quite precocious and read encyclopedias and other books while performing sophisticated science experiments in his room, always preferring to work in solitude. Also at a young age Nash read ET Bell's famous book, Men of Mathematics, and was particularly taken with the chapter on Pierre de Fermat - the lawyer and number theorist - which instigated Nash's interest in mathematics.

Young John attended public schools, but was also tutored by his mother outside of the classroom, which helped him later 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 algebraic geometry as well, writing a paper titled, "The Imbedding Problem For Riemannian Manifolds," which solved the long-standing problem of the same name.

Nash taught advanced math courses as an instructor at M.I.T. for awhile 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 the 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 Princeton University and would wander around the library, whistling, reading, and also writing nonsensical equations and other phrases on various black boards, 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 he was headed to Europe. While there, he sent a variety of postcards to friends and family as he traveled around hoping to legally give up his U.S. citizenship. On one of the postcards, Nash mentioned a number he dubbed, 'The Beautiful Buddhist number.' 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 that Nash gave this number, it sounds as if it could have supernatural or mystical properties. But we must also keep in mind that he was also suffering from his mental condition when he discovered it. 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 the mathematical world. Also the numbers 22 and 4 make an appearance. Why did Nash choose 22 and 4? Well, 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 which 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? Maybe so. Only John Nash could tell us 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. So my questions to the reader are: where do you think five 5s will occur? And do you think anyone will ever find a large number, such as, say, nine hundred and ninety nine 999s in a row in the decimal expansion? (I realize these questions are not especially serious mathematics, but that they fall more into the realm of campy mathematics, but that is all right with me - I like campy things in life.) 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? Although we need a real mathematician to answer these questions, I will say that I think the Beautiful Buddhist number is worthy of further study and I'm quite intrigued with its formulation and would like to know more about the number, so I hope some real mathematicians tackle 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, which I'm proud of. The description of the sequence runs: "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 Encyclopeida of Integer Sequences database, if you're interested.

Now I will end this article with a quote from John Forbes Nash Jr. - "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."

And here are the first 500 digits of the Beautiful Buddhist number for your contemplation.

22 * Pi + 4 * e =

79.988165692811632187619304317559713443366715161052166361316651541668267459710788348102294871627241205317123471020222985923334235796628983389559244159868967106858779573954421427375247501227889789472479263729178216373344431183064000679979266615012284673533327672990037316844964568030443089516036461076300147725192693367985172931792264190423325323979206520585440723086586508583825251659374020851550495974298607224148771457597883183027412076597148540422844721314159060362269199306183899333562196946658053...

Sources:

Red Zen, Jason Earls, Pleroma Publications, 2007.

John Nash, Wikipedia, http://en.wikipedia.org/wiki/John_Forbes_Nash

22 (number), Wikipedia, http://en.wikipedia.org/wiki/22_(number)