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Remark. x3 + c, c ∈ Z, this so-called tangent For Bachet’s equation y2 method gives an interesting duplication formula.

M. Edwards’s articles on Kronecker in History and Philosophy of Modern Mathematics (Minneapolis, MN, 1985) 139–144, Minnesota Stud. Philos. , XI, Univ. ) In calculus, the field of rational numbers Q is insufficient for several reasons, including convergence. Thus, Q is extended to the field of real numbers R. It is visualized by passing the Cartesian Bridge2 connecting algebra and geometry; each real number (in infinite decimal representation) corresponds to a single point on the real line.

Rationality, Elliptic Curves, and Fermat’s Last Theorem Hammurabi were well aware of the significance of these numbers, including the fact that they are integral side lengths of a right triangle. The time scale is quite stunning; this is about 1000 years before Pythagoras! The first infinite sequence of all Pythagorean triples (a, b, c) with b,c consecutive was found by the Pythagoreans themselves: (a, b, c) (n, (n2 − 1)/2, (n2 + 1)/2), where n > 1 is odd. Plato found Pythagorean triples (a, b, c) with b + 2 c, namely, (4n, 4n2 − 1, 4n2 + 1), n ∈ N.

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A New Proof of the Lefschetz Formula on Invariant Points by Hopf H.

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