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Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
6
votes
Accepted
Rational solutions to x^3 + y^3 + z^3 - 3xyz = 1
I think this surface has a rational parameterization in terms of (a,b), given by:
$x = (1 + a + a^2)^2/(9 (3 + a (6 + (-1 + a)^2 a)) b^2) + ((-2 + (-2 + a) a) b)/(1 + a + a^2)$
$y = (1 + a + a^2)^2 …
11
votes
Accepted
Geometry of numbers for three by three matrices?
The volume of $S_1$ is $\frac{16767 \pi^4}{560} \approx 2916.53$.
The idea is to realize that the set $S_1$ can be equivalently described as the nuclear norm (sum of the singular values $\sigma_i$) o …