I came across a Web paper entitled The Impact of Google’s Random Number Generator on Accurate Results that I found most puzzling in failing to explain the specific nature of this random generator and that included gems as below
“PRNGs (…) outputs are inherently predictable given a specific seed value, thus deviating from true randomness”
or stating that linear congruential generators suffered from “discernible patterns”, or yet advancing a bonus in using Google’s generator for “leveraging Gaussian distributions”… Referring to a specific and possibly obscure arXival proposing to construct (yet) a PRNG by reinforced learning, with an objective function based on randomness tests reminded me of Marsaglia’s Die Hard. And to “true randomness”. Plus being repetitive and obsequious. Until I found the piece was “written” by Quthor with no typo, an AI “writer” powered by Quick Creator. I should have paid more attention to the level of nonsense…
On the same occurrence, I also read a recent Scientific American paper on randomness tests pseudo-random number generators by Christopher Lutsko. Which does not start that well, since it reproduces the Laplace démon’s argument that a die roll outcome is not random. Also repeating the above pleonasm that PRNGs are not random since they are deterministic. And failing to not mention lava generators! The core of the article is however about recent papers by the author and Niclas Technau of the only examples of sequences that prove passed extremely strong pseudorandom tests. They focus on tests based on gap distribution and pair correlation.
“The [gap distribution] measures the size of the gaps between the points, and the [pair correlation’ measures the clustering of the points—how much they group up or stay apart (…) If these agree with what we would expect from random data, we say that the gap distribution or pair correlation is “Poissonian”.” Christopher Lutsko
Along with Athanasios Sourmelidis, they proved that exponential sequences {α exp(θ log[n]) mod 1} have Poissonian pair correlation when θ<⅓, for any value of α. And higher Poissonian correlations for θ “smaller and smaller“. (This does not come out of the blue, as the proof relates to Van der Corput’s method of exponential sums, with links to the leading Austrian random generator community.) This is definitely an achievement. However, when θ gets “smaller and smaller“, the terms in the sequence grow more and more slowly, which forces calling for larger and larger values of α to make the generator useful. And the pseudo-randomness test based on Poissonian correlations is only one of many from the toolbox, so the debate seems far from over.
‘s into uniforms 
and look at the d dimensional vectors
, which gives 41 hyperplanes when 
… So in this sense all generators share the same poor property as the infamous RANDU which is such that that 
is always over one of 16 hyperplanes, an exercise we use in both 
‘s share 
‘s, but of course the subsequence 
also 
