Notes 33
so that j(τ) 1/q + 744 for small q. Now for the coup de grˆ ace: One
can show that if K is an imaginary quadratic field with integral basis 1,τ,
then j(τ) is an algebraic integer of degree exactly h(K), the class number
of K. In particular, if K has class number 1, then is a rational integer.
The main theorem of §8 implies that K = Q(
√j(τ)
−163) has class number
1, and so j(τ) Z for τ =
1+i

163
2
. This value of τ corresponds to q =
−1/ exp(π

163), so that


163
j(τ) 744 Z.
We remark that


163
is actually transcendental, as may be deduced from
the following theorem of Gelfond and Schneider (noting that


163
=
(−1)i

163):
If α and β are algebraic numbers, where α = 0 and β is irra-
tional, then
αβ
is transcendental. Here
“αβ”
stands for exp(β log α), and any
nonzero value of log α is permissible. For a proof of the Gelfond–Schneider
result, see, e.g., [Hua82, §17.9].
There are many sequences not discussed in §10 where it would be of
interest to decide if they contain infinitely many primes, or composites.
For example, fix a nonintegral rational number α 1, and consider the
sequence of numbers
αn
. Whiteman has conjectured that this sequence
always contains infinitely many primes. If we drop the rationality condi-
tion, then from a very general theorem of Harman [Har97] we have that
each sequence
αn
contains infinitely many primes as long as α 1 avoids
a set of measure zero. (Of course since the rational numbers have measure
zero, this has no direct consequence for Whiteman’s conjecture.) Very little
is known about the sequences considered by Whiteman. For the particular
numbers α = 3/2 and α = 4/3, Forman & Shapiro [FS67] present ingenious
elementary arguments showing that the sequence
αn
contains infinitely
many composite numbers. Some extensions of their results have been ob-
tained by Dubickas & Novikas [DN05]; e.g., these authors prove that if
ξ 0 and α {2, 3, 4, 6, 3/2, 4/3, 5/4}, then the sequence
ξαn
contains
infinitely many composites.
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