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**Title**: Variations on a Theme of Euler**Author**: Takashi Ono**Category**: Mathematics**Date**: March 2013**Lenguage**: English**Pages**: unknown**ISBN**: 1475723261

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The first six chapters and Appendix 1 of this guide appeared in Japanese in a guide of the same title 15years aga (Jikkyo, Tokyo, 1980).At the request of some individuals who don’t want to be taught Japanese, I made a decision to rewrite my previous work in English. This time, I added a chapter on the arithmetic of quadratic maps (Chapter 7) and Appendix 2, A Short Survey of Subsequent Research on Congruent Numbers, by M. Kida. Some 20 years ago, whereas rifling by the pages of Selecta Heinz Hopj (Springer, 1964), I observed a system of three quadratic types in 4 variables with coefficientsin Z that yields the map of the 3-sphere to the 2-sphere with the Hopf invariant r =1 (cf. Selecta, p. 52). Immediately I feit that one side of classical and trendy quantity idea, together with quadratic types (Pythagoras, Fermat, Euler, and Gauss) and area elliptic curves as intersection of quadratic surfaces (Fibonacci, Fermat, and Euler), could possibly be thought of because the quantity idea of quadratic maps-especially of these maps sending the n-sphere to the m-sphere, i.e., the generalized Hopf maps. Having these in thoughts, I deliveredseverallectures at The Johns Hopkins University (Topics in Number Theory, 1973-1974, 1975-1976, 1978-1979, and 1979-1980). These lectures essentially contained the next three primary areas of arithmetic: v vi Preface Theta Simple Functions Aigebras Elliptic Curves Number Theory Figure P.l

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