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... Science Frontiers ONLINE No. 42: Nov-Dec 1985 Issue Contents Other pages Home Page Science Frontiers Online All Issues This Issue Sourcebook Project Sourcebook Subjects The Fabric Of Prime Number Distribution Mathematicians and many non-mathematicians have soft spots in their hearts for numbers that cannot be subdivided; that is, the prime numbers. No one has ever been able to figure out any foolproof system to their occurrence or how to generate them all by formula. Some primes, such as 11 and 13, 17 and 19, etc., come in pairs; but most don't . And some formulas are particularly good at generating primes, but they all fail somewhere. One such formula was discovered by Leonhard Euler, the great mathematician: n2+ n+ 41 This formula works for n ... 0, 1, 2,...39; but fails at n = 40. An interesting consequence of Euler's formula can be made apparent when all numbers from 41 to 440 are written in a square spiral, like so: All of the numbers on the diagonal indicated are Euler formula primes, even when the spiral is expanded to 20 x 20. However, when the 20 x 20 spiral is examined closely, many of the other primes -- those not generated by Euler's formula -- also tend to line up on diagonals. This is a most intriguing characteristic, one which goes far beyond the 20 x 20 array mentioned above. The computer-generated display shown below lays out a huge square spiral, with each prime a bright dot. The ...
Terms matched: 2 - Score: 740 - 15 May 2017 - URL: /sf042/sf042p25.htm
... Project Sourcebook Subjects 1, 089, 533, 431, 247, 059, 310, 875, 780, 378, 922, 957, 732, 908, 036, 492, 993, 138, 195, 385, 213, 105, 561, 742, 150, 447, 308, 967, 213, 141, 717, 486, 151 This 97-digit number is a prime, divisible by only 1 and itself. But, add 210 to it, and you get still another prime. Add another 210, and another prime pops up! You can do this six times and gets a series of seven consecutive primes in an arithmetic progression. Neat! And just a tiny bit of order in the distribution of primes. It took H. Dubner and H. ... This Issue Sourcebook Project Sourcebook Subjects 1, 089, 533, 431, 247, 059, 310, 875, 780, 378, 922, 957, 732, 908, 036, 492, 993, 138, 195, 385, 213, 105, 561, 742, 150, 447, 308, 967, 213, 141, 717, 486, 151 This 97-digit number is a prime, divisible by only 1 and itself. But, add 210 to it, and you get still another prime. Add another 210, and another prime pops up! You can do this six times and gets a series of seven consecutive primes in an arithmetic progression. Neat! And just a tiny bit of order in the distribution of primes. It took H. Dubner ...
Terms matched: 2 - Score: 643 - 15 May 2017 - URL: /sf102/sf102m20.htm
... Science Frontiers ONLINE No. 111: May-Jun 1997 Issue Contents Other pages Home Page Science Frontiers Online All Issues This Issue Sourcebook Project Sourcebook Subjects Something Strange Is Going On!Where? "Everywhere, of course," is the answer of any anomalist worth his or her salt. Especially, though, something strange in going on with prime numbers. In an homage to the revered mathematician P. Erdos, who died September 20, 1996, D. Mackenzie mentioned a theory Erdos published in 1940 with M. Kac. This theory states that a plot of the number of prime factors of very large numbers forms a bell curve -- almost as if these numbers were "choosing" their prime factors at random. Alluding to a assertion Einstein is said to have made, ... commented: "God may not play dice with the universe, but something strange is going on with the prime numbers." (Mackenzie, Dana; "Homage to an Itinerant Master," Science, 275:759, 1997.) Cross reference. The distribution of prime numbers is more than strange, see the plot in SF#42/332. What do prime numbers have to do with the real world? Are math and natural science really separate, unlinked disciplines? Pythagoras, 2,500 years ago, decided that: "All is number." He may be right. A strange connection seems to exist between prime numbers and quantum physics. On one side of the chasm that supposedly separates math from physics, we have the prime numbers and the Riemann zeta function ...
Terms matched: 2 - Score: 576 - 15 May 2017 - URL: /sf111/sf111p00.htm
... multiple of 11. Moreover, similar relationships occur where the interval between the chosen integers is a power of 5, 7, or 11 or a product of these powers. Ramanujan was able to prove that these curious patterns also hold for all higher numbers beyond 200. Ramanujan's discovery came as quite a surprise to the world of mathematics, as did the strange roles of the three adjacent prime numbers 5, 7, and 11. Recently, though, K. Ono has gone beyond Ramanujan and proved that there are really an infinite number of relationships like the three found by Ramanujan. (Peterson, Ivars; "The Power of Partitions," Science News, 157:396, 2000.) From Science Frontiers #132, NOV-DEC 2000 . 2000 William R. ... . Ramanujan's has been called a "magical genius" because his remarkable insights seemed to come out of the blue -- like magic. We have not neglected Ramanujan in this newsletter ( SF#53 and here ), and now we spotlight him again. First, a quick primer on a fascinating mathematical byway called "partitions." A partition is a way in which a whole number can be expressed as the sum of positive integers. For example, 5 can be partitioned in seven ways: 5 4+ 1 3+ 2 3+ 1+ 1 2+ 2+ 1 2+ 1+ 1+ 1 1+ 1+ 1+ 1+ 1 The number 4 has only five partitions. Check it out. Historically, ordinary mortals saw no patterns in ...
Terms matched: 2 - Score: 332 - 15 May 2017 - URL: /sf132/sf132p12.htm
... ONLINE No. 87: May-Jun 1993 Issue Contents Other pages Home Page Science Frontiers Online All Issues This Issue Sourcebook Project Sourcebook Subjects Calculating prodigies, gnats, and smart weapons In a thought-provoking letter to New Scientist, J. Margolis commences with the observation that calculating prodigies (idiot savants), who are often also mentally retarded, can easily and almost instantaneously recognize 20-digit prime numbers! Gifted mathematicians with so-called photographic memories cannot perform such mental feats using known methods for identifying primes. What do the calculating prodigies know that the rest of us do not? Better algorithms; that is, calculating methods? Margolis expands on this: "All this suggests some relatively simple, subconscious algorithms which have not, as yet, been explicitly formulated. Research in this ... might well result in new mathematical insights. "It need not be surprising that mathematical insight is more fundamental than language. Even a primitive animal brain is 'wired" to perform exceedingly complex computations essential for survival in an unpredictable environment. The latest 'smart' weapons are rudimentary compared with a humble gnat. Mathematics could be a by-product of these functions. Language is a comparatively recent evolutionary innovation and it is quite possible that conscious manipulation of abstract symbols has not caught up with an innate ability to perceive quantitative relationships." (Margolis, Joel; "What Gnats Know," New Scientist, p. 58, January 30, 1993.) From Science Frontiers #87, MAY-JUN 1993 . 1993-2000 William R. Corliss ...
Terms matched: 2 - Score: 303 - 15 May 2017 - URL: /sf087/sf087p17.htm
... Manufactures Phony Blueberry Flowers Music in the Ear Guiding Cell Migration Remarkable Distribution of Hydrothermal Vent Animals Trees May Not Converse After All! Geology Feathers Fly Over Fossil 'Fraud' Sand Dunes 3 Kilometers Down The Night of the Polar Dinosaur Geophysics The Sausalito Hum Mysterious Hums: the Sequel Psychology Left-handers Have Larger Interbrain Connections Geomagnetic Activity and Paranormal Experiences Taking Food From Thought Logic & Mathematics The Fabric of Prime Number Distribution Chemistry & Physics Speculations From Gold ...
Terms matched: 2 - Score: 270 - 15 May 2017 - URL: /sf042/index.htm
... ' time on Earth Stealth fish Geology The Dwarfing of island megafauna and the remarkable survival of some A double-whammy for the Yucatan, but that's only part of the story Geophysics A sign? Star-of-David ice crystals fall upon West Sussex Hessdalen: Valley of enigmatic lights When coming events really cast their shadows before them! Physics Entangled moments Mathematics Patterns of very loosely knit prime numbers ...
Terms matched: 2 - Score: 270 - 15 May 2017 - URL: /sf155/index.htm
... R. Highfield in the Chicago Sun-Times . "These savants are often autistic, a developmental disorder that leaves them with little ability to empathize with others. However, some possess astonishing skills. "He [Snyder] believes the ability to tap raw information -- the mind's secret arithmetic -- is possessed by mathematical savants. They can multiply, divide, factor and identify prime numbers of six and more digits in seconds, or identify the number of objects they can see at a single glance -- 111 matches scattered on the floor, in one case." Snyder's intriguing conclusion is that ". .. we believe that everyone has the underlying facility to perform lightningfast integer arithmetic." (Highfield, Roger; "Study Adds Up to Formula for ... Genius," Chicago Sun Times , March 23, 1999. Cr. J. Cieciel) A more technical review of the SnyderMitchel work has appeared in Nature. There, N. Birbaumer focussed on that mysterious barrier that supposedly prevents most of us from utilizing our innate genius. Unfortunately, his explanations are a bit murky and jargony. We normal people cannot use our innate talents "because we process information in a concept-driven way." Savants, however, can tap these capabilities because of "a functional or pathological loss of executive brain centres." In other words, the way we are programmed to think blocks or suppresses access to our reservoir of mathematical talents. In his review, Birbaumer adds that the work of Snyder and Mitchel is contradicted by studies of non-savant ...
Terms matched: 2 - Score: 261 - 15 May 2017 - URL: /sf125/sf125p00.htm
