Prime Numbers

24 Chapter 1 PRIMES!
9.5.18 and Algorithm 9.5.19). This method has at least doubled the search
efficiency over previous methods.
It should be mentioned that modern Mersenne searching is sometimes of
the “hit or miss” variety; that is, random prime exponents q are used to check
accordingly random candidates 2 q − 1. (In fact, some Mersenne primes were
indeed found out of order, as indicated above). But much systematic testing
has also occurred. As of this writing, exponents q have been checked for all
q ≤ 12830000. Many of these exponents are recognized as giving composite
Mersennes because a prime factor is detected. For example, if q is a prime
that is 3 (mod 4), and p =2q + 1 is prime, then p|Mq. (See also Exercises
1.47, 1.81.) For the remaining values of q, the Lucas–Lehmer test (see Section
4.2.1) was used. In fact, for all q ≤ 9040000 for which a factor of Mq was
not found, the Lucas–Lehmer test was carried out twice (see [Woltman 2000],
which website is frequently updated).
As mentioned in Section 1.1.2, the prime M25964951 is the current record
holder as not only the largest known Mersenne prime, but also the largest
explicit number that has ever been proved prime. With few exceptions, the
record for largest proved prime in the modern era has always been a Mersenne
prime. One of the exceptions occurred in 1989, when the “Amdahl Six” found
the prime [Caldwell 1999]
391581 · 2 216193 − 1,
which is larger than 2 216091 − 1, the record Mersenne prime of that time.
However, this is not the largest known explicit non-Mersenne prime, for Young
found, in 1997, the prime 5·2 240937 +1, and in 2001, Cosgrave found the prime
3 · 2 916773 +1.
Actually, the 5th largest known explicit prime is the non-Mersenne
5359 · 2 5054502 +1,
found by R. Sundquist in 2003.
Mersenne primes figure uniquely in the ancient subject of perfect numbers.
A perfect number is a positive integer equal to the sum of its divisors other
than itself. For example, 6 = 1 + 2 + 3 and 28 = 1 + 2 + 4 + 7 + 14 are
perfect numbers. An equivalent way to define “perfection” is to denote by
σ(n) the sum of the positive divisors of n, whence n is perfect if and only if
σ(n) =2n. The following famous theorem completely characterizes the even
perfect numbers.
Theorem 1.3.3 (Euclid–Euler). An even number n is perfect if and only if
it is of the form
n =2 q−1 Mq,
where Mq =2 q − 1 is prime.