Richard Feynman and 1/243
by Jason Earls, author of How to Become a Guitar Player from Hell & Heartless Bastard In Ecstasy http://becomeguitaristfromhell.blogspot.com/ http://www.youtube.com/user/zevi35711
When the great physicist Richard Feynman was at Los Alamos in the 1940s working on building the first atomic bomb (now known as the Manhattan Project), he did a lot of calculating with the ‘human computers’ group, and one day noticed the following intriguing pattern in the decimal expansion of 1/243. 1/243 = 0.004115226337448559670781893004115226337448559670781893 004115226337448559670781893004115226337448559670781893 00411522633744855967078189300411... Feynman liked the pattern so much he mentioned it to his wife in a letter. But the good people monitoring the correspondence to and from Los Alamos did not like seeing the decimal expansion of 1/243 appear in his missive, since they thought he might be a spy sending a dangerous coded message. So Feynman had numerous arguments with the Los Alamos officials to get everything straightened out with the 1/243 letter and others thereafter. The decimal expansion of 1/243 is just as interesting to me as it was to Richard Feynman (who was awarded the Nobel Prize in 1965 for his work on quantum electrodynamics), perhaps even moreso, thus I decided to investigate the number a little more and the first thing I wondered was if it contained any primes in its decimal expansion. (I also looked for squares, triangulars, fibonaccis, and harshad numbers, like a postmodern experimental mathematician wasting away in a doomed and abandoned metropolis.) I typed and diagrammed and racked my brain and bit my lip for hours – no, that’s not really true. What I programmed was actually fairly simple and I quickly discovered that this number
411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189300 411522633744855967078189300411522633744855967078189299
is prime; meaning that it has no divisors except for itself and one. The number above is actually the first 810 digits of the decimal expansion of 1/243 (disregarding the first two zeros) after subtracting 1. If we let RF(n) = floor(1/243 * 10n+2 ) – 1, then RF(n) is prime when n = 1, 158, and 810, with no more found up to 1000. I have dubbed primes of this type “Atomic Bomb 243” primes. Note that I found no numbers n such that RF(n) is a square and no numbers n such that RF(n) is triangular, but I did find that when n = 1, 2, 3, 18, 105, and 268, RF(n) is a Harshad number with no more up to 1000. I knew I would probably find a few Harshads, but squares and
triangular numbers in decimal expansions are much more challenging to find. (The RF(810) prime above originally appeared in my novel, Cocoon of Terror, which was published by Afterbirth Books. Help me out by purchasing a copy today!)
Do you think it is asinine to look for different classes of numbers in the decimal expansion of 1/243? I don’t think so. Not at all. I consider it interesting and provocative, enchanting and noble, even as explosive as an atomic bomb if it gets inside the right bloke’s mind. I only wish I could find more classes of numbers among the digits of 1/243. Notice that the main pattern in the decimal expansion of 1/243 can be broken down into two main chunks: 004115226337448559 and 670781893 The first section is where the pleasing symmetry comes in: 004115226337448559, while the next section, 670781893, is not symmetrical since the pattern is disturbed from “carries” involved with multiplication. What other properties might the two numbers above have? Well, 004115226337448559 has many prime factors (notice below the two initial zeros were dropped): 4115226337448559 = 3*31*61*283*491*5220511 and 670781893 is a semiprime with only two prime factors: 670781893 = 4549 * 147457 Unfortunately those are the only properties I can discern regarding these numbers. Perhaps you can find something more interesting. I showed the large prime above to my wife and told her the story of Feynman and his wife and the letter and 243. But she did not comment on it. She only nodded and smiled. Maybe she didn’t understand what I was telling her. She doesn’t seem to care about mathematics or numbers very much. She seems to loathe any kind of math and I think she even despises computers, but that doesn’t really matter. In my opinion, looking for “Atomic Bomb 243” primes is not decadent or stupid. Finding bizarre numerical tidbits that are legitimate mathematical facts is interesting and worthwhile. And sometimes when I am staring at the interesting numbers I find, they remind me of throngs and chiffres and jillions of estimates floating over a freemason’s halo with drizzle falling from his soft eyelashes as he dreams mystical thoughts, or of a friendly milkman wearing a green wig with fish eggs sliding from his kneecaps as he hides treasures in a dry town to escape rapid-fire tumbleweeds so he can become a new mathematical messiah and free everyone from their private suffering and individual humiliations. Only kidding. Remember. One divided by two hundred and fourty three is 0.0041152263374485596707818930041152263374485596707818 ... And that’s all anybody really needs to know.
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(Thanks for reading. If you have any comments or know 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://www.youtube.com/user/zevi35711 http://becomeguitaristfromhell.blogspot.com/ http://zombiesofthereddescent.blogspot.com/ Bio: Jason Earls is the author of Cocoon of Terror (Afterbirth Books), How to Become a Guitar Player from Hell, Heartless Bastard In Ecstasy, 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, MathWorld.com, three of Clifford Pickover’s books, Scientia Magna, 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.