ZX Computings - OpenLibra

Dr. Frank O 'Hara at home in
Surrey proudly holding a ZX81
printout of the largest known
prime number: 2 ' *44497 -1.lt
was discovered by Harry Nelson,
47, and David Slowinski. 25, in
1979, at the Lawrence Livermore
Laboratory in California, after a
two-month run on a Cray One
computer. It has 13395 decimal
digits. It took just over two and a
half hours to calculate these digits
on a ZX81, using a machine code
program. The printout, which is
seven feet, one and a haff inches
long, took 15 minutes to produce
on the ZX Printer.
32
Mathematics
Delving
numerically
deeper
Frank O'Hara from Surbiton in
Surrey helped lan Logan decode
the 8K ROM. Dr O'Hara has
continued his investigations
into the operation of tne ZX81,
and here shares with us some
of his discoveries, with notes
on some programs on
elementary number theory for
the ZX81.
Over the past year or so, Dr
O'Hara has developed a few
programs on elementary
number theory for the ZX81,
having previously run a few
such programs on a programmable
calculator, a Texas Tl
58, over about 2 Vi years from
mid-1 978.
A couple of these programs
are "one off", ie. ad hoc pro
grams to solve a single program.
There is one of about
100 bytes of machine code
which generates the decimal
representation of quite large
powers of 2. He used this to obtain
the 13395 digits of (2 to
the 44497} minus 1, the
largest known prime number,
discovered by Nelson and
Slowinski using a Cray One
computer in 1979. The program
took 2 hours 31 minutes
to obtain this number on the
ZX81.
Another even more exciting
result was given by about 400
bytes of machine code, including
a multiple precision
multiplication routine. This program
actually proved the
primality of the first 1 5
Mersenne primes, up to and including
(2 to the 1 279) minus
1, a number of 386 digits. It
thus repeated some of the work
of "SWAC" in 1 953, going far
beyond what the desk calcu
lators had done and capturing
some of the flavour of a historic
moment (although 28 years
later!).
Three other programs have a
more general purpose flavour, I
have called them:
(a) SPRF: single precision prime
factorization;
(b) MPRF: multiple precision
prime factorization;
(c) FE24: Fermat's theorem used
to test numbers up to 24
digits long for compositeness,
ie. lack of primality.
The kernel of these 3 programs
is the machine code
multiple precision integer division
routine which finds the
true integer quotient and true
integer remainder of an integer
of arbitrary length with respect
to another arbitrarily long integer.
Barden is quite mistaken
when he describes this process
as a "cop-out" (How to Program
Microcomputers, by
William Barden, Jr. Sams, Indianapolis,
1 977, page 109). It
is in fact very easy to program.
It is a simple extension of the
standard restoring division of
one or two bytes by one byte.
The shift is just a loop with its
kernel as: LD A, (DE): RLA : LD
{DE), A. The addition or subtraction
is another loop centred
on: LD A, (DE) : ADC A, (HL> :
LD (DE), A; and so on. In fact
the only complexity arises
when one has to shorten the
process in order to speed it up,
as in the first of the 3 programs,
SPRF. This program has a 4
byte dividend and 2 byte
divisor, and needs to use the exchange
resisters H', L\ D' and
E' to gain speed.
The first program, SPRF,
finds the smallest prime factor
of any odd number from 5 to
4294967255 (2 to the 32
minus 1) in not more than 20
seconds. If the numberis prime,
the program reports this. It
goes about 1000 times as fast
as the Texas calculator did.
This has been achieved by a
series of improvements, starting
with a BASIC program that
was only about 10 times as fast
as the calculator. The program
contains about 400 bytes of
machine code; 300 or so of
these are just a simple linear
sieve designed to exclude
multiples of 3, 5 and 7 as well
as 2 and so gain a factor of
35/16 in speed. The speed has
to be seen to be believed.
Numbers up to 7 digits long are
dealt with instantaneously. The
largest 8. 9 and 1 0 digit primes
in its range take 3, 10 and 20
seconds respectively. The program
can easily be adapted to
print screenfuls of results and
so, for example, find the largest
prime less than 2 to the 32 in
one run. By using random 8, 9
or 10 digit input one can use it
to see primes probably never
seen before. (Only the first 10
or 1 1 million numbers are completely
and accurately listed as
prime or composite.)
ZX COMPUTING AUG/SEPT 1982 28