An experimental exploration of Marsaglia's xorshift generators, scrambled

Marsaglia proposed recently xorshift generators as a class of very fast, good-quality pseudorandom number generators. Subsequent analysis by Panneton and L'Ecuyer has lowered the expectations raised by Marsaglia's paper, showing several weaknesses of such generators, verified experimentally using the TestU01 suite. Nonetheless, many of the weaknesses of xorshift generators fade away if their result is scrambled by a non-linear operation (as originally suggested by Marsaglia). In this paper we explore the space of possible generators obtained by multiplying the result of a xorshift generator by a suitable constant. We sample generators at 100 equispaced points of their state space and obtain detailed statistics that lead us to choices of parameters that improve on the current ones. We then explore for the first time the space of high-dimensional xorshift generators, following another suggestion in Marsaglia's paper, finding choices of parameters providing periods of length $2^{1024} - 1$ and $2^{4096} - 1$. The resulting generators are of extremely high quality, faster than current similar alternatives, and generate long-period sequences passing strong statistical tests using only eight logical operations, one addition and one multiplication by a constant

Sums of two biquadrates and elliptic curves of rank ≥4\geq 4

If an integer $n$ is written as a sum of two biquadrates in two different ways, then the elliptic curve $y^2=x^3-nx$ has rank $\geq 3$. If moreover $n$ is odd and the parity conjecture is true, then it has even rank $\geq 4$. Finally, some examples of ranks equal to 4, 5, 6, 7, 8 and 10, are also obtained.Comment: 11 pages, 2 table

A curve algebraically but not rationally uniformized by radicals

Zariski proved the general complex projective curve of genus g>6 is not rationally uniformized by radicals, that is, admits no map to the projective line whose Galois group is solvable. We give an example of a genus 7 complex projective curve Z that is not rationally uniformized by radicals, but such that there is a finite covering Z' -> Z with Z' rationally uniformized by radicals. The curve providing the example appears in a paper by Debarre and Fahlaoui where a construction is given to show the Brill Noether loci W_d(C) in the Jacobian of a curve C may contain translates of abelian subvarieties not arising from maps from C to other curves.Comment: 8 pages, AMSlate