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The mathematical constant e can be represented in a variety of ways as a real number.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.
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
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 ...
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 ...
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
For the continued fraction expansion. of a number, see simple continued fraction, of a function, see continued fraction This page was last edited on 11 ...
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His proofs are similar to Fourier's proof of the irrationality of e. In 1891, Hurwitz explained how it is possible to prove along the same line of ideas that e is not a root of a third-degree polynomial with rational coefficients, which implies that e 3 is irrational. [12] More generally, e q is irrational for any non-zero rational q. [13]