Prime Numbers

76 Chapter 1 PRIMES!
These numbers are “probable primes” (see Chapter 3). True primality proofs
have not been achieved (and these examples may well be out of reach, for the
foreseeable future!).
1.81. Candidates Mp = 2 p − 1 for Mersenne primes are often ruled out
in practice by finding an actual nontrivial prime factor. Work out software
for finding factors for Mersenne numbers, with a view to the very largest
ones accessible today. You would use the known form of any factor of Mp
and sequentially search over candidates. You should be able to ascertain, for
example, that
460401322803353 | 2 20295923 − 1.
On the issue of such large Mersenne numbers; see Exercise 1.82.
1.82. In the numerically accessible region of 2 20000000 there has been at least
one attempt at a compositeness proof, using not a search for factors but the
Lucas–Lehmer primality test. The result (unverified as yet) by G. Spence is
that 2 20295631 − 1 is composite. As of this writing, that would be a “genuine”
composite, in that no explicit proper factor is known. One may notice that
this giant Mersenne number is even larger than F24, the latter recently having
been shown composite. However, the F24 result was carefully verified with
independent runs and so might be said still to be the largest “genuine”
composite.
These ruminations bring us to a research problem. Note first a curious
dilemma, that this “game of genuine composites” can lead one to trivial claims,
as pointed out by L. Washington to [Lenstra 1991]. Indeed, if C be proven
composite, then 2 C −1, 2 2C −1 −1 and so on are automatically composite. So in
absence of new knowledge about factors of numbers in this chain, the idea of
“largest genuine composite” is a dubious one. Second, observe that if C ≡ 3
(mod 4) and 2C +1 happens to be prime, then this prime is a factor of 2 C −1.
Such a C could conceivably be a genuine composite (i.e., no factors known) yet
the next member of the chain, namely 2 C − 1, would have an explicit factor.
Now for the research problem at hand: Find and prove composite some number
C ≡ 3 (mod 4) such that nobody knows any factors of C (nor is it easy to
find them), you also have proof that 2C + 1 is prime, so you also know thus
an explicit factor of 2 C − 1. The difficult part of this is to be able to prove
primality of 2C + 1 without recourse to the factorization of C. Thismightbe
accomplished via the methods of Chapter 4 using a factorization of C +1.
1.83. Though it is unknown whether there are infinitely many Mersenne
or Fermat primes, some results are known for other special number classes.
Denote the n-thCullennumberbyCn = n2 n + 1. The Cullen and related
numbers provide fertile ground for various research initiatives.
One research direction is computational: to attempt the discovery of prime
Cullen numbers, perhaps by developing first a rigorous primality test for the
Cullen numbers. Similar tasks pertain to the Sierpiński numbers described
below.