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In this case, both x and y are odd and z is even. Since (y 2, z, x 2) form a primitive Pythagorean triple, they can be written z = 2de y 2 = d 2 − e 2 x 2 = d 2 + e 2. where d and e are coprime and d > e > 0. Thus, x 2 y 2 = d 4 − e 4. which produces another solution (d, e, xy) that is smaller (0 < d < x). As before, there must be a lower ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
The first four partial sums of the series 1 + 2 + 3 + 4 + ⋯.The parabola is their smoothed asymptote; its y-intercept is −1/8, and the area of the parabola ...
Let X be the subvariety of the four-dimensional affine plane, with coordinates x,y,z,w, generated by y 2-x 3 and x 4 +xz 2-w 3. The canonical desingularization of the ideal with these generators would blow up the center C 0 given by x=y=z=w=0. The transform of the ideal in the x-chart if generated by x-y 2 and y 2 (y 2 +z 2-w 3).
The case (x, y, z) = (3, 5, 5) and all its permutations were proven by Bjorn Poonen in 1998. [25] The case (x, y, z) = (3, 6, n) and all its permutations were proven for n ≥ 3 by Bennett, Chen, Dahmen and Yazdani in 2014. [5] The case (x, y, z) = (2n, 3, 4) and all its permutations were proven for n ≥ 2 by Bennett, Chen, Dahmen and Yazdani ...