Prime Numbers

1.3 Primes of special form 25
Proof. Suppose n =2 a m is an even number, where m is the largest odd
divisor of n. The divisors of n are of the form 2 j d, where 0 ≤ j ≤ a
and d|m. Let D be the sum of the divisors of m excluding m, and let
M =2 a+1 − 1=2 0 +2 1 + ···+2 a . Thus, the sum of all such divisors of
n is M(D + m). If M is prime and M = m, thenD = 1, and the sum of all
the divisors of n is M(1 + m) =2n, sothatn is perfect. This proves the first
half of the assertion. For the second, assume that n =2 a m is perfect. Then
M(D + m) =2n =2 a+1 m =(M +1)m. Subtracting Mm from this equation,
we see that
m = MD.
If D>1, then D and 1 are distinct divisors of m less than m, contradicting
the definition of D. SoD =1,m is therefore prime, and m = M =2 a+1 − 1.
✷
The first half of this theorem was proved by Euclid, while the second half
was proved some two millennia later by Euler. It is evident that every
newly discovered Mersenne prime immediately generates a new (even) perfect
number. On the other hand, it is still not known whether there are any odd
perfect numbers, the conventional belief being that none exist. Much of the
research in this area is manifestly computational: It is known that if an odd
perfect number n exists, then n>10 300 , a result in [Brent et al. 1993], and that
n has at least eight distinct prime factors, an independent result of E. Chein
and P. Hagis; see [Ribenboim 1996]. For more on perfect numbers, see Exercise
1.30.
There are many interesting open problems concerning Mersenne primes.
We do not know whether there are infinitely many such primes. We do not
even know whether infinitely many Mersenne numbers Mq with q prime
are composite. However, the latter assertion follows from the prime k-tuples
Conjecture 1.2.1. Indeed, it is easy to see that if q ≡ 3 (mod 4) is prime and
2q + 1 is also prime, then 2q + 1 divides Mq. For example, 23 divides M11.
Conjecture 1.2.1 implies that there are infinitely many such primes q.
Various interesting conjectures have been made in regard to Mersenne
numbers, for example the “new Mersenne conjecture” of P. Bateman,
J. Selfridge, and S. Wagstaff, Jr. This stems from Mersenne’s original assertion
in 1644 that the exponents q for which 2 q −1 is prime and 29 ≤ q ≤ 257 are 31,
67, 127, and 257. (The smaller exponents were known at that time, and it was
also known that 2 37 −1 is composite.) Considering that the numerical evidence
below 29 was that every prime except 11 and 23 works, it is rather amazing
that Mersenne would assert such a sparse sequence for the exponents. He was
right on the sparsity, and on the exponents 31 and 127, but he missed 61, 89,
and 107. With just five mistakes, no one really knows how Mersenne effected
such a claim. However, it was noticed that the odd Mersenne exponents below
29 are all either 1 away from a power of 2, or 3 away from a power of 4 (while
the two missing primes, 11 and 23, do not have this property), and Mersenne’s
list just continues this pattern (perhaps with 61 being an “experimental error,”
since Mersenne left it out). In [Bateman et al. 1989] the authors suggest a new