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March 2000 Pebbles to Microchips. A History of Algorithms: From the Pebble to the Microchip, edited by JeanLuc Chabert, is reviewed by Jeremy Gray in the February 14 2000 Nature. An algorithm is a recipe for solving a certain kind of problem. Many mathematical formulas are in fact algorithms, since they specify a sequence of operations to be performed on their "inputs." One of the best known ones is Beyond Quadratic Reciprocity. Ian Porteous picks up an item by Jonathan Rogawsky from the January 2000 AMS Notices in the January 15 Science News. Quadratic reciprocity is an amazing law of number theory discovered by Gauss. In its simplest form it relates facts about arithmetic modulo p and arithmetic modulo q, where p and q are primes greater than 2, and says that p is a perfect square mod q if and only if q is/is not a perfect square mod p. "Is" unless both p and q give remainder 3 when divided by 4, "is not" when they both do. For example 13 and 17 (in the "is" category): The perfect squares mod 13 are 1, 4, 9, 3, 12 and 10; the prefect squares mod 17 are 1, 4, 9, 16, 8, 2, 15, and 13. Notice that 17 (=4 mod 13) is in the mod 13 list and 13 is in the mod 17 list. This general phenomenon is still considered ``one of the deepest and most mysterious results of elementary number theory.'' The quote is from Rogawsky, whose article concerns the recent proof of the ``local Langlands correspondence,'' a farreaching generalization of Gauss' discovery.
Tony Phillips 
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