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Morning Edition for April 10, 2009, carried a piece by Joe Palca ("Mining for the 'Prime' Jewels of Numbers") about the search for "the world's largest prime number." A serious possible misunderstanding is averted when, at the end of the next paragraph, he specifies: "But there's always a larger one to find." This is after telling us about the latest contender in words we can understand: "if you write 10 digits per inch--all 12,978,189 of them--the number would extend for 20.45 miles." Palca mentions that the recent largest primes have been Mersenne primes and that the "current reigning champ ... was discovered last summer as part of a program called the Great Internet Mersenne Prime Search, or GIMPS." Palca asked Chris Caldwell (UTM), one of the mathematicians behind this gigantic distributed computing effort, why it was worth the trouble. Cadwell answers with an analogy: when he went to Washington, he took his kids to see the Hope Diamond at the National Museum of Natural History. "Mersennes, in a way, are kind of like a large diamond. Nobody there looking at the Hope Diamond ever asks, 'Why did they bother to dig it up?' or 'What is it good for?' - even though it really isn't good for much other than to just hang there and people to look at. And in many ways the Mersennes play that same role - that they really are the jewels of number theory."
Some of the gaps in Palca's presentation (e.g. what is a Mersenne prime?) are filled in a companion piece ("See The Largest Known Prime, All 13 Million Digits") by Andrew Price, on the same NPR webpage. But neither cites Euclid or mentions that 211-1 = 2047 = 89 x 23, elementary but meaningful in this context.
A double pendulum with labels for lengths, masses, angular cordinates and velocities; the motion-tracking data fed into Schmidt and Lipson's algorithm (color codes match the diagram); the conserved quantity the algorithm detected: the system's Hamiltonian. Adapted from an image kindly provided by Hod Lipson.
Recently two Cornell scientists have found an algorithmic way to "identify and document analytical laws that underlie physical phenomena in nature." As Michael Schmidt and Hod Lipson describe their work in Science (April 3, 2009), "A key challenge to finding analytic relations automatically is defining algorithmically what makes a correlation in observed data important and insightful." They propose what they call "a principle for the identification of nontriviality," and exhibit its application to the analysis of motion-tracking data from various physical systems; for example the double pendulum illustrated above, where the conserved quantity detected by their algorithm is the system's Hamiltonian. To test the significance ("non-triviality") of a quantity f(x,y), detected by their algorithm to be constant, their idea is to measure the discrepancy between the implicit derivative δy/δx = (∂f/∂x)/(∂f/∂y) calculated from f, and the implicit derivative Δy/Δx = (dy/dt)/(dx/dt) calculated from the continuing stream of data. "In higher-dimensional systems, multiple variable pairings and higher-order derivatives yield a plethora of criteria to use."
Schmidt and Lipson give several examples of how reasonable their algorithm is. In the case of the double pendulum, when the algorithm was only given data measured in high-energy runs, it "fixated" on angular momentum, which is conserved (to good approximation) in that context. On the other hand, "given only data from low-velocity in-phase oscillations, the algorithm fixated on small-angle approximations and uncoupled energy terms." Finally, "By combining the chaotic data with low-velocity in-phase oscillation data, the algorithm converged onto the precise energy laws after several hours of computation." The paper's title is "Distilling Free-Form Natural Laws from Experimental Data."
Part I of this story is tucked inside a long profile of the famous physicist in the March 25 2009 New York Times Magazine (The article, by Nicholas Dawidoff, stirred up a firestorm of outraged commentary because it allowed Dyson to present his iconoclastic views on global warming). Dyson participates in Jason, a small super-classified think-tank the government runs "each summer near San Diego." At lunch, one of the scientists "will idly wonder if there is an integer where, if you take its last digit and move it to the front, turning, say, 112 to 211, it's possible to exactly double the value. Dyson will immediately say, 'Oh, that's not difficult,' allow two short beats to pass and then add, 'but of course the smallest such number is 18 digits long.'" The meal ends in silence with nobody having "the slightest idea how Freeman could have known such a fact or, even more terrifying, could have derived it in his head in about two seconds." (This last quote from William Press, who presumably was there).
Part II is in the online New York Times, in two installments of the TierneyLab, a science blog on the website. On April 6, John Tierny posts the Dawidoff quote, along with an analysis from Pradeep Mutalik (Medical Informatics, Yale). "In fact, the procedure to find the answer requires no more than 4th-grade arithmetic skills. I know, because I actually showed my fourth-grader daughter, Maya, how to do it, and she had no problem whatsoever in computing the answer." Mutalik calls the number in Dawidoff's account the Dyson number for 2, generalizes the problem to finding the Dyson number for n (moving the last digit to first place multiplies the original number by n), defines "reverse Dyson numbers," etc.
On April 10, Tierney publishes Mutalik's 4-th grade daughter's solution. (Meanwhile he had heard from Freeman Dyson: "I am sure I had seen the problem before .... Mr. Dawidoff made a big deal out of something very ordinary. The problem is well known among recreational mathematicians.") As instructed by her father, she started with a 2 on the right side of a sheet of paper, and extended towards the left as follows. The second digit is twice the first, the third is twice the second, etc., with "carries" added in as she went along. She stopped when she found a number starting with 10....
2 42 842 16842 136842 736842 14736842 94736842 1894736842 17894736842 157894736842 1157894736842 3157894736842 63157894736842 1263157894736842 5263157894736842 105263157894736842
Moving the last 2 to the front gives 210526315789473684, manifestly twice 105263157894736842. The April 10 blog also contains Dr. Mutalik's explanation of the phenomenon in terms of arithmetic mod 19.
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