42 1. Elementary Prime Number Theory, I
25. (Schinzel [Sch62a]) In 1857, Bunyakovsky conjectured [Bun57] that if
F (T) Z[T] is an irreducible polynomial with positive leading coeffi-
cient and D is the largest positive integer dividing F (n) for each n Z,
then F (n)/D is prime for infinitely many natural numbers n. Show that
this would follow from Hypothesis H.
26. (Granville; see, e.g., [Mol97, Theorem 2.1]) Assume Hypothesis H.
Show that for every natural number N0, one can find a positive in-
teger A with the property that
n2
+ n + A assumes prime values for all
0 n N0. Hint: Apply Hypothesis H to the N0 linear polynomials
T, T +
(12
+ 1),T +
(22
+ 2),...,T + (N0
2
+ N0).
27. (Schinzel & Sierpi´ nski [SS58]) Assume Hypothesis H. Show that if n 1
and r is a positive integer divisible by all primes p n, then there
are infinitely many arithmetic progressions of length n and common
difference r consisting of consecutive primes.
Remark. The weaker claim that there are arbitrarily long arithmetic
progressions of primes was recently proved in a technical tour de force
by Green & Tao [GT08], using ideas borrowed from ergodic theory (and
several other fields). For some striking elementary consequences of the
Green–Tao result, see [Gra08a].
28. (Cf. Chang & Lih [CL77]) Show that for every N N, there is a
polynomial F (T) Z[T] for which {F (k)}k=0
N
is a sequence of N + 1
distinct primes. Hint: For 0 k N, put ck(T) =
0≤i≤N,i=k
(T
i). Using Corollary 1.20, choose integers r0,r1,...,rN for which {1 +
rkck(k)}k=0
N
is a sequence of N + 1 distinct primes. Put F (T) := 1 +
∑N
i=0
rici(T).
29. (Clement [Cle49], Cucurezeanu [Cuc68]) Let k and n be integers with
n k 2. Suppose that n has no prime divisors k. Show that n and
n + k are simultaneously prime if and only if
k · k!((n 1)! + 1) + (k!
(−1)k)n
0 (mod n(n + k)).
30. (Shanks [Sha64]) Let F (z) =
∑∞
n=0
zn(n+1)/2
and define
G(z) := (F (z)
1)2
(F (z) 1).
Prove that there are infinitely many primes of the form
n2+1
2
(with
n N) if and only if the power series expansion of G has infinitely
many negative coefficients.
31. Suppose p 3 (mod 4) is prime. Prove that if 2p +1 is also prime, then
2p + 1 |
2p
1. Deduce that Hypothesis H implies Conjecture 1.27.
32. (Selfridge; cf. [Erd50b]) Let n N. Show that
78557·2n
+1 is divisible
by some prime number from the set {3, 5, 7, 13, 19, 37, 73} . In particular,
78557 ·
2n
+ 1 is always composite.
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