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

eπ

√

163

≈ j(τ) − 744 ∈ Z.

We remark that

eπ

√

163

is actually transcendental, as may be deduced from

the following theorem of Gelfond and Schneider (noting that

eπ

√

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.