Discrete Time Signal Processing by A.W.M.Van Den Enden, N.A.M. Verhoeckx

By A.W.M.Van Den Enden, N.A.M. Verhoeckx

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Elliptic curve | one branch Figure 5. Elliptic curve | two branches 28 I. C/. Our primary interest is in examining the group of rational points that is defined below. C/ can be made into Lie groups. e. the complex torus. This leads to another way of looking at elliptic curves, since the complex torus is also isomorphic to C/L, where L is a lattice in the complex plane. This viewpoint has been fundamental in the development of the theory of elliptic functions; see the text by Akhiezer [1, Chapter 1].

It must be noted that the description of the above solution as using a method of tangents is of course anachronistic. It is not a valid historical interpretation of the methods of Diophantus; rather it is a way of describing his method that connects his work to the later tradition of Fermat, Euler, Cauchy, and Lucas. Viete and Fermat extended the algebraic methods of Diophantus (see the surveys by Bashmakova [3], Norio [36], and Weil [48, Chapter 2]). It should be noted that all of these approaches implicitly used the tangent line at a point on a cubic.

We are interested in his work on Diophantine equations, but he is perhaps best known for his work on 32 I. Algebra, Number Theory, Calculus, and Dynamical Systems primality testing. He originated the procedure now known as the Lucas{Lehmer test for primes. Using this method, Lucas showed that the Mersenne number 2127 1 is prime. This remains the largest prime number found without use of a computer. Lucas also did considerable work in recreational mathematics. For example, he invented the Tower of Hanoi puzzle (well-known to computer science students).

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