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If (x, y) is a rational point on a Mordell curve with y ≠ 0, then so is ( x 4 − 8nx / 4y 2 , −x 6 − 20nx 3 + 8n 2 / 8y 3 ). Moreover, if xy ≠ 0 and n is not 1 or −432, an infinite number of rational solutions can be generated this way.
Then let x and y be ordered 5-element vectors containing the above data: x = (1, 2, 3, 5, 8) and y = (0.11, 0.12, 0.13, 0.15, 0.18). By the usual procedure for finding the angle θ between two vectors (see dot product ), the uncentered correlation coefficient is
"Rational Solutions to x^y = y^x". CTK Wiki Math. Archived from the original on 2021-08-15 "x^y = y^x - commuting powers". Arithmetical and Analytical Puzzles. Torsten Sillke. Archived from the original on 2015-12-28. dborkovitz (2012-01-29). "Parametric Graph of x^y=y^x". GeoGebra.
Whether a faster algorithm exists is an open problem, [1] [2] posed by Elwyn Berlekamp prior to 1975. [1] [3] A variant of the problem sorts the sumset, the set of sums of pairs, with duplicate sums condensed to a single value.
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
The points (x,y,z) of the sphere x 2 + y 2 + z 2 = 1, satisfying the condition x = 0.5, are a circle y 2 + z 2 = 0.75 of radius on the plane x = 0.5. The inequality y ≤ 0.75 holds on an arc. The length of the arc is 5/6 of the length of the circle, which is why the conditional probability is equal to 5/6.
[1] The fifth problem on Hilbert's list is a generalisation of this equation. Functions where there exists a real number c {\displaystyle c} such that f ( c x ) ≠ c f ( x ) {\displaystyle f(cx)\neq cf(x)} are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of Hilbert's third problem ...
The entry 4+2i = −i(1+i) 2 (2+i), for example, could also be written as 4+2i= (1+i) 2 (1−2i). The entries in the table resolve this ambiguity by the following convention: the factors are primes in the right complex half plane with absolute value of the real part larger than or equal to the absolute value of the imaginary part.