Pythagoras and the Pythagoreans
13
take a short detour to detail some of the modern methods employed in
the search. Though this is a departure from ancient Greek mathematics,
the contrast and similarity between then and now is remarkable. Just
the fact of finding perfect numbers using the previous propositions has
spawned a cottage industry of determining those numbers
p
for which
2
p
−
1
is prime. We call a prime number a
Mersenne Prime
if it has the
form
2
p
−
1
for some positive integer
p
. Named after the friar
Marin
Mersenne
(1588 - 1648), an active mathematician and contemporary
of Fermat, Mersenne primes are among the largest primes known today.
So far 38 have been found, though it is unknown if there are others
between the 36th and 38th. It is not known if there are an infinity of
Mersenne primes. From Euclid’s theorem above, we also know exactly
38 perfect numbers. It is relatively routine to show that if
2
p
−
1
is
prime, then so also is
p
.
18
Thus the known primes, say to more than
ten digits, can be used to search for primes of millions of digits.
Below you will find complete list of Mersenne primes as of January,
1998. A special method, called the
Lucas-Lehmer
test has been devel-
oped to check the primality the Mersenne numbers.
18
If
p
=
rs
, then 2
p
−
1 = 2
rs
−
1 = (2
r
)
s
−
1 = (2
r
−
1)((2
r
)
s
−
1
+ (2
r
)
s
−
2
· · ·
+ 1)