Mathematics of Random Number Generation

Question

I am interested in learning more about Random Number Generation. Recently, I found out about an algorithm called "RANDU" (https://en.wikipedia.org/wiki/RANDU), which was an early algorithm for generating random numbers.

Using the R language, I implemented it below:

v = rep(0,100)
v[1]=3
for (j in 1:100) {
v[j+1]=(65539*v[j])%%(2^31)
}

    i = 1:101

range01 <- function(x, ...){(x - min(x, ...)) / (max(x, ...) - min(x, ...))}
      
    rand_data = data.frame(i,v)
    rand_data$int_version = range01(rand_data$v)

 head(rand_data)
  i        v  int_version
1 1        3 0.000000e+00
2 2   196617 9.201486e-05
3 3  1179675 5.520835e-04
4 4  5308497 2.484362e-03
5 5 21233907 9.937413e-03
6 6 79626969 3.726522e-02
    

plot(rand_data$v, type = "b", main = "100 Random Real Numbers with Randu")

Naturally, this lead me to the following question:

• How do we know that these 100 numbers are truly random - Is there some mathematical proof for this?
• Why does this RANDU algorithm work - Is there some mathematical proof for this?

I tried to look into this, but I couldn't find a straight answer. Can someone please provide any comments?

Thank you!

Note: Apparently you can use the Chi-Square Test to find out if a random sequence of numbers has a "uniform distribution" (https://stats.stackexchange.com/questions/406406/test-of-uniform-distribution-using-ks-test-and-chi-square-in-r, Is there a simple test for uniform distributions? ... but I am not sure why the Chi-Square Test can detect if random numbers have a uniform distribution). I performed this test in R:

ks.test(rand_data$int_version, "punif", min(rand_data$int_version), max(rand_data$int_version))

    One-sample Kolmogorov-Smirnov test

data:  rand_data$int_version
D = 0.079507, p-value = 0.5457
alternative hypothesis: two-sided

Based on these results, the p-value is significantly greater than a standard tolerance level (e.g. alpha = 0.05) - therefore, there is sufficient statistical evidence to accept the Null Hypothesis and conclude that these random numbers correspond to a uniform distribution.