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An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA).
The number of points (n), chords (c) and regions (r G) for first 6 terms of Moser's circle problem. In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem (named after Leo Moser), has a solution by an inductive method.
The number of steps to calculate the GCD of two natural numbers, a and b, may be denoted by T(a, b). [96] If g is the GCD of a and b, then a = mg and b = ng for two coprime numbers m and n. Then T(a, b) = T(m, n) as may be seen by dividing all the steps in the Euclidean algorithm by g. [97]
The above two steps are repeated for each multiplier digit. At the end, the result appears in the accumulator windows. In this way, the operand can be multiplied by as large a number as desired, although the result is limited by the capacity of the accumulator. To divide by a multidigit divisor, this process is used:
To jump ahead k steps on each iteration (using the f function from that section), break up the current number into two parts, b (the k least significant bits, interpreted as an integer), and a (the rest of the bits as an integer). The result of jumping ahead k is given by f k (2 k a + b) = 3 c(b, k) a + d(b, k).
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The average number of steps it performs is r 2. [citation needed] This fact is the discrete version of the fact that a Wiener process walk is a fractal of Hausdorff dimension 2. [citation needed] In two dimensions, the average number of points the same random walk has on the boundary of its trajectory is r 4/3.
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