40 | New Scientist | 10 August 2019
There is also prize money at stake. Pace
received $3000 for his discovery, which he
gave to his church. But future prime hunters
could earn a lot more. The Electronic Frontier
Foundation, a technology-focused non-profit,
is behind the $150,000 prize for anyone who
discovers a prime with at least 100 million
digits. It has also pledged $250,000 for a prime
with over a billion digits.
The main reason Mersenne primes are
attractive, however, is that they tend to be the
largest primes we can find. That is because
there is a particularly efficient way to test if
a Mersenne number is prime. It was devised
by French mathematician Édouard Lucas, who
in 1857, at the age of 15, used it to test whether
the Mersenne number 2^127 -1 was prime.
If you have a number like 7, the obvious
way to check if it is prime is to look at whether
it can be divided by the numbers below it. With
huge numbers, however, that is a ridiculously
laborious process. Lucas flipped it on its head.
He identified a sequence of numbers with a
remarkable property: if a Mersenne number
can divide into another, larger number in that
sequence, then the Mersenne is prime. This
means you have one long calculation to make,
instead of lots and lots of them.
Even so, the task was far from trivial. Lucas
pulled it off by representing the Mersenne
number in binary form on a 127 by 127
chessboard, with pawns for the 1s and empty
squares for the 0s. By moving the pawnsSPLM77232917 is only
the 50th Mersenne
prime ever foundcomputers grew faster and more powerful, the
primes we discovered grew longer (see “Prime
targets”, right). It wasn’t until George Woltman
launched the collaborative GIMPS project in
1996 that the general public could get involved.
A programmer and prime lover, Woltman
wanted everyone to have a chance at
discovering giant primes. He rendered the
algorithm so fast and efficient that it can run
in the background on any relatively modern
household computer with an internet
connection. Then he worked with others
to automate the selection and allocation
of numbers to test, and the checking and
reporting of results.
The fruit of this is a program so easy to use
that hundreds of people have signed up for a
stab at finding a record-breaking prime (see
“How to become a number hunter”, page 39).
“From the 1950s through to ’96, you had to
own a supercomputer to have a chance to play
the game,” says Woltman. “Not anymore.”
The only thing you need, beyond a computer,
is patience. The software will run on most
machines, but on a typical home computer
it takes roughly 14 days to test a potential
Mersenne prime.Number crunchers
Among the first to sign up to GIMPS was Curtis
Cooper, a mathematician at the University of
Central Missouri, who has been fascinated by
primes since childhood. He wasn’t messing
about. While many volunteers donate the
spare processing power of their own computer,
as the administrator for a whole campus,
Cooper was able to recruit some 600 machines.
That gave him a better chance than most. Surearound, he could painstakingly carry out the
division. It took him 19 years, presumably not
full time. But eventually, Lucas had his result:
M127 is indeed prime.
No one ever reproduced Lucas’s efforts.
Indeed, he himself only performed the entire
operation once. But US mathematician
Derrick Henry Lehmer refined the method
in the 1930s to create the Lucas-Lehmer
test. It remains the simplest way to test if a
Mersenne number is prime, and it formed
the basis of the algorithms that brought the
search for primes into the computer age. AsIn November 2016, after
105 days of furious
computation, Peter Trueb’s
machine spat out a gigantic
number. It was pi, the
famous mathematical
constant that roughly
equals 3.14, but here it
had been calculated to
record-breaking precision:
some 22.4 trillion digits
after the decimal point.
A software developer in
Zurich, Switzerland, Trueb
is one of hundreds of
people across the world
who use a freely available
computer program called
y-cruncher to find record
numbers of digits forvarious mathematical
constants, from pi and
Euler’s number to the
golden ratio.
This form of number
exploration relies heavily
on computer power. Trueb,
a lifelong pi enthusiast
who is also interested in
the philosophy of numbers,
used his company’s
resources to build a
computer with 24 hard
drives. Each contained
6 terabytes of memory,
to store the whopping
quantity of data generated
by the calculations.
He was always likely to
be outgunned eventually,however. Sure enough, his
record was smashed this
year by a Google developer
from Japan, Emma Haruka
Iwao, who used the tech
giant’s cloud computing
services to calculate pi to
31.4 trillion digits.
There is more serendipity
involved in other forms of
number exploration. Rather
than pinning down the
digits of a special number
to extreme precision,
these involve interrogating
figures lurking in the
furthest reaches of the
number line to find those
that are the rarest and most
beautiful (see main story).CONSTANT CRAVINGS
We’re all mathematicians
Bobby Seagull explains everyday maths at New Scientist Live
newscientistlive.com/maths