So each has a fixed number of possible values and it will simply cycle through those in the exact same order each time if you actually call it 2**$bits times. The number of effectively different seed values is called the "period".

Sorry tye, but you are very wrong on this, and I can prove it.

The "period" is the number of values you have to draw before the entire sequence of 2**n-bits values, repeat in the same sequence, in their entirety!

The Mersenne Twister MT19937, with 32-bit word length, has a period of 2¹⁹⁹³⁷ - 1. How could that be if your statement above was true?

Of course, that is not a LCPRNG.

But, the rand() provided by MSVC is a linear congruential PRNG. And it only produce 15-bits on entropy. Proof:

C:\test>perl -E"++$h{ int( rand 65536 ) } for 1 .. 1e6; say scalar key
+s %h; grep{ $_ & 1 } keys %h or say 'No odd numbers found'"
32768
No odd numbers found
[download]

And for the whole sequence to repeat, the first two values would have to repeat one after the other first. And according to you, that should happen within the first 32768 values produced.

It doesn't. (With srand(1)):

C:\test>randperiod -M=2
41 18467
i:1 n:412286284
First sequence of 2 values repeated itself after 412286284 calls to ra
+nd
[download]

That's 412 million values generated before the first pair are repeated in sequence.

Now let's try to match the first 3 values produced:

C:\test>randperiod -M=3
41 18467 6334
i:2 n:2147418117
First sequence of 3 values repeated itself after 2147418117 calls to r
+and
[download]

That's 2 billion values drawn before the first 3 values repeat in sequence.

The test code:

#! perl -slw
use strict;

sub rand32768{ int( rand 32768 ) }

$|++;
our $M //= 10;

srand( 1 );

my @first = map rand32768(), 1 .. $M;
print "@first";

my $n = $M;

OUTER: while( 1 ) {
    ++$n until rand32768 == $first[ 0 ];
    for my $i ( 1 .. $M - 1 ) {
        ++$n;
        printf "\ri:$i n:$n";
        redo OUTER unless rand32768() == $first[ $i ];
    }
    last;
}

print "\nFirst sequence of $M values repeated itself after $n calls to
+ rand";
[download]

I've tweaked the code to log how many value were drawn for each increasing length sequence. I'll leave it running over night.