30 1. Elementary Prime Number Theory, I
It is now known that Fm is composite for 5 m 32, and (for the same
probabilistic reasons alluded to above) it is widely believed that Fm is com-
posite for every m 5. So much for intuition! Despite this widespread
belief, the following conjecture appears intractable:
Conjecture 1.28. The Fermat number Fm is composite for infinitely many
natural numbers m.
Similarly, for each even natural number a, one can look for primes in
the sequence
a2m
+1. Again we believe that there should be at most finitely
many, but again the analogue of Conjecture 1.28 seems impossibly difficult!
Indeed, there is no specific even number a for which we can prove that
a2m
+1 is composite infinitely often. This is a somewhat odd state of affairs
in view of the following amusing theorem of Schinzel [Sch63]:
Theorem 1.29. Suppose that infinitely many of the Fermat numbers Fj are
prime. If a 1 is an integer not of the form
22r
(where r 0), then
a2m
+1
is composite for infinitely many natural numbers m.
Proof. Fix an integer a 1 not of the form
22r
. Let M0 be an arbitrary
positive integer. We will show that
a2m
+ 1 is composite for some m M0.
Let Fj be a prime Fermat number not dividing
a(a2M0
1). Since a is
coprime to Fj, Fermat’s little theorem implies that
aFj −1
=
a22j
1 (mod Fj).
Since Fj
a2M0
1, we must have M0
2j.
So we can write
aFj −1
1 =
a22
j
1
=
(a2M0

1)(a2M0
+
1)(a2M0+1
+
1)(a2M0+2
+ 1) · · ·
(a22j
−1
+ 1).
Since Fj divides
aFj −1
1 but not
a2M0
1, it must be that Fj divides
a2m
+ 1 for some M0 m
2j.
We cannot have
a2m
+ 1 = Fj, since a is
not of the form
22r
, and so
a2m
+ 1 is composite.
In connection with Fermat-type numbers the following result of Shapiro
& Sparer [SS72] merits attention (cf. [Sha83, Theorem 5.1.5]). It shows
(in particular) that the doubly exponential sequences
a2m
+ 1 are unusually
difficult to handle among sequences of the same general shape:
Theorem 1.30. Suppose a, b, and c are integers, and that a, b 1. If c
is odd, then
abm
+ c
is composite for infinitely many m N, except possibly in the case when a
is even, c = 1, and b =
2k
for some k 1. If c is even, there are infinitely
many such m except possibly when a is odd and c = 2.
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