Quick HP42S question

[Albert Chan]

(05-25-2020 02:35 PM)Valentin Albillo Wrote:              N     N^th random number
     ---------------------------------
            1     0.731362440213
           10     0.723555832003
          100     0.056136887198
        1 000     0.411709251980
       10 000     0.823130519800
      100 000     0.935105198000
    1 000 000     0.831051980000
   10 000 000     0.410519800000
  100 000 000     0.205198000000

Not very random-looking, methinks, and that many trailing zeros ...

This apply to any LCG RNG: seed * a mod m

k-th random number = seed * a^k mod m

If k = order(a,m), we have a^k mod m = 1
If k | order(a,m), we expect a non-random pattern for s = a^k mod m

Example, to reproduce above numbers:

>>> a, m = 2851130928467, 10**15
>>> seed = 10**14 + 11       # seed for RANDOMIZE 1
>>> for i in xrange(9):      # note: order(a,m) = 5E13
...     n = 10 ** i          # note: n | order(a,m)
...     s = pow(a,n,m)
...     print '%-9d %015d %015d' % (n, s, s*seed % m)
...
1         002851130928467  731362440213137
10        165777802909449  723555832003939
100       268739717018001  056136887198011
1000      573791750180001  411709251980011
10000     429375501800001  823130519800011
100000    439555018000001  935105198000011
1000000   975550180000001  831051980000011
10000000  755501800000001  410519800000011
100000000 555018000000001  205198000000011

Most significant digits are more random than least siginificant digits.
Thus, last 3 digits were chopped off, to produce RND()