Prime Numbers

1.6 Research problems 77
A good, simple exercise is to prove that there are infinitely many composite
Cullen numbers, by analyzing say Cp−1 for odd primes p.Inadifferentvein,Cn
is divisible by 3 whenever n ≡ 1, 2(mod6)andCn is divisible by 5 whenever
n ≡ 3, 4, 6, 17 (mod 20). In general show there are p−1 residue classes modulo
p(p − 1) for n where Cn is divisible by the prime p. It can be shown via sieve
methods that the set of integers n for which Cn is composite has asymptotic
density 1 [Hooley 1976].
For another class where something, at least, is known, consider Sierpiński
numbers, being numbers k such that k2 n + 1 is composite for every positive
integer n. Sierpiński proved that there are infinitely many such k. Prove
this Sierpiński theorem, and in fact show, as Sierpiński did, that there is
an infinite arithmetic progression of integers k such that k2 n + 1 is composite
for all positive integers n. Every Sierpiński number known is a member of
such an infinite arithmetic progression. For example, the smallest known
Sierpiński number, k = 78557, is in an infinite arithmetic progression of
Sierpiński numbers; perhaps you would enjoy finding such a progression. It
is an interesting open problem in computational number theory to decide
whether 78557 actually is the smallest. (Erdős and Odlyzko have shown on
the other side that there is a set of odd numbers k of positive asymptotic
density such that for each k in the set, there is at least one number n with
k2 n + 1 prime; see [Guy 1994].)
1.84. Initiate a machine search for a large prime of the form n = k2 q ± 1,
alternatively a twin-prime pair using both + and −. Assume the exponent q
is fixed and that k runs through small values. You wish to eliminate various k
values for which n is clearly composite. First, describe precisely how various
values of k could be eliminated by sieving, using a sieving base consisting of
odd primes p ≤ B, whereB is a fixed bound. Second, answer this important
practical question: If k survives the sieve, what is now the conditional heuristic
“probability” that n is prime?
Note that in Chapter 3 there is material useful for the practical task
of optimizing such prime searching. One wants to find the best tradeoff
between sieving out k values and actually invoking a primality test on the
remaining candidates k2 q ± 1. Note also that under certain conditions on the
q, k, there are relatively rapid, deterministic means for establishing primality
(see Chapter 4).
1.85. The study of prime n-tuplets can be interesting and challenging. Prove
the easy result that there exists only one prime triplet {p, p +2,p +4}.
Then specify a pattern in the form {p, p + a, p + b} for fixed a, b such that
there should be infinitely many such triplets, and describe an algorithm for
efficiently finding triplets. One possibility is the pattern (a =2,b = 6), for
which the starting prime
p =2 3456 + 5661177712051
gives a prime triplet, as found by T. Forbes in 1995 with primalities proved
in 1998 by F. Morain [Forbes 1999].