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Suppose that the public keys are N, e = 90581, 17993 . The attack should determine d. By using Wiener's theorem and continued fractions to approximate d, first we try to find the continued fractions expansion of e / N . Note that this algorithm finds fractions in their lowest terms. We know that
Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus, e may also be represented as an infinite series, infinite product, or other types of limit of a sequence.
Lagrange's discovery implies that the canonical continued fraction expansion of the square root of every non-square integer is periodic and that, if the period is of length p > 1, it contains a palindromic string of length p − 1. In 1813 Gauss derived from complex-valued hypergeometric functions what is now called Gauss's continued fractions ...
The terms in the continued fraction expansion of every Liouville number are unbounded; using a counting argument, one can then show that there must be uncountably many transcendental numbers which are not Liouville. Using the explicit continued fraction expansion of e, one can show that e is an example of a transcendental number that is not ...
Euler derived the formula as connecting a finite sum of products with a finite continued fraction. (+ (+ (+))) = + + + + = + + + +The identity is easily established by induction on n, and is therefore applicable in the limit: if the expression on the left is extended to represent a convergent infinite series, the expression on the right can also be extended to represent a convergent infinite ...
Download QR code; Print/export Download as PDF; ... For the continued fraction expansion. of a number, see simple continued fraction, of a function, see ...
Continued fraction for ζ(3/2) Jan 1, 2003: A080000: Number of permutations satisfying −k ≤ p(i) − i ≤ r and p(i) − i: Feb 10, 2003: A090000: Length of longest contiguous block of 1s in binary expansion of nth prime. Nov 20, 2003: A091345: Exponential convolution of A069321(n) with itself, where we set A069321(0) = 0. Jan 1, 2004: A100000
A simple or regular continued fraction is a continued fraction with numerators all equal one, and denominators built from a sequence {} of integer numbers. The sequence can be finite or infinite, resulting in a finite (or terminated) continued fraction like