If S is a set of positive integers which contains 1, 2, …, n–1, but not n or any multiple of n, where n ≥ 2, then

Here R is the field of real numbers, and ∥α∥ denotes the distance of α from the nearest integer.

Mr V.E. Hoggatt, Jr, has pointed out an error in the examples of my paper [2], If Fm denotes the mth Fibonacci number, these examples asserted that

is transcendental. This is in fact false, for by a theorem of Good [1],

for it happens that

is a rational and not a transcendental function of z, so that Theorem 1 of my paper cannot be applied. The value of s follows from (1) on putting .

Hence the following changes have to be made in [2].

On p. 390, lines 7 and 10, the case k = 1 must each time be excluded, and in Theorem 2 the two numbers r and s may not be both be 0.

The proof of Lemma 2 in [1] invoked elementary analytic number-theory. I have just realized that there is a proof which is entirely elementary. It is doubtless “well-known” (in the usual technical sense that it appears somewhere in the literature) and it is certainly “well-knowable” in Conway's terminology. However, as it renders the entire argument of my paper elementary, I give it here.