<s>
RANDU	B-Algorithm
is	O
a	O
linear	B-Algorithm
congruential	I-Algorithm
pseudorandom	B-Algorithm
number	I-Algorithm
generator	I-Algorithm
(	O
LCG	B-Algorithm
)	O
of	O
the	O
Park	B-Algorithm
–	I-Algorithm
Miller	I-Algorithm
type	I-Algorithm
,	O
which	O
was	O
used	O
primarily	O
in	O
the	O
1960s	O
and	O
1970s	O
.	O
</s>
<s>
IBM	O
's	O
RANDU	B-Algorithm
is	O
widely	O
considered	O
to	O
be	O
one	O
of	O
the	O
most	O
ill-conceived	O
random	O
number	O
generators	O
ever	O
designed	O
,	O
and	O
was	O
described	O
as	O
"	O
truly	O
horrible	O
"	O
by	O
Donald	O
Knuth	O
.	O
</s>
<s>
It	O
fails	O
the	O
spectral	B-Algorithm
test	I-Algorithm
badly	O
for	O
dimensions	O
greater	O
than	O
2	O
as	O
will	O
be	O
seen	O
below	O
.	O
</s>
<s>
For	O
any	O
linear	B-Algorithm
congruential	I-Algorithm
generator	I-Algorithm
with	O
modulus	O
m	O
used	O
to	O
generate	O
points	O
in	O
n	O
-	O
dimensional	O
space	O
,	O
the	O
points	O
fall	O
in	O
no	O
more	O
than	O
parallel	O
hyperplanes	O
.	O
</s>
<s>
This	O
indicates	O
that	O
low-modulus	O
LCGs	B-Algorithm
are	O
unsuited	O
to	O
high-dimensional	O
Monte	B-Algorithm
Carlo	I-Algorithm
simulation	I-Algorithm
.	O
</s>
<s>
For	O
m	O
=	O
2^31	O
and	O
n	O
=	O
3	O
,	O
an	O
LCG	B-Algorithm
could	O
have	O
up	O
to	O
2344	O
planes	O
,	O
theoretical	O
maximum	O
.	O
</s>
<s>
Now	O
we	O
examine	O
the	O
values	O
of	O
multiplier	O
65539	O
and	O
modulus	O
231	O
chosen	O
for	O
RANDU	B-Algorithm
.	O
</s>
<s>
Even	O
by	O
the	O
standards	O
of	O
LCGs	B-Algorithm
,	O
this	O
shows	O
that	O
RANDU	B-Algorithm
is	O
terrible	O
:	O
using	O
RANDU	B-Algorithm
for	O
sampling	O
a	O
unit	O
cube	O
will	O
only	O
sample	O
15	O
parallel	O
planes	O
,	O
not	O
even	O
close	O
to	O
the	O
upper	O
limit	O
of	O
2344	O
planes	O
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
wide	O
use	O
of	O
RANDU	B-Algorithm
in	O
the	O
early	O
1970s	O
,	O
many	O
results	O
from	O
that	O
time	O
are	O
seen	O
as	O
suspicious	O
.	O
</s>
<s>
This	O
misbehavior	O
was	O
already	O
detected	O
in	O
1963	O
on	O
a	O
36-bit	O
computer	O
,	O
and	O
carefully	O
reimplemented	O
on	O
the	O
32-bit	O
IBM	B-Application
System/360	I-Application
.	O
</s>
<s>
The	O
start	O
of	O
the	O
RANDU	B-Algorithm
's	O
output	O
period	O
for	O
the	O
initial	O
seed	O
is	O
:	O
</s>
