Park City Mathematics Institute 2003
Number Theory - Bowen Kerins, Ben Sinwell, and Brian Hopkins
Sums and Differences: the Art and Craft of Adding and Subtracting: How do you find a polynomial function of smallest degree that agrees with a table of data? What's the sum of the fourth powers of all the integers between 1 and 100? What's the sum of the reciprocals of the perfect squares? What's the sum of the squares of all the complex numbers whose fifth power is 1? Why, if the third differences in an input-output table are constant, is there a cubic polynomial fit? What's the probability that two integers, chosen at random, have no common factor? And, most importantly, what do all these questions have to do with one another? This course looks at the calculus of finite differences as a unifying theme for these questions and others that connect to the middle and high school curriculum.
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IAS/Park City Mathematics Institute is an outreach program of the Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540.
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With generous support from Robert and Lynn Johnston