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Fix relatively prime integers $a>b>0$ and an integer $n>2$. Let $P(n)$ be the greatest prime factor of $n$, and let $\Phi_n=\Phi_n(a,b)$ be the homogeneous $n$th cyclotomic factor.

Statement

For $n>2$, $P(n)^2\nmid\Phi_n$, and each prime factor of $\Phi_n$ other than $P(n)$ is congruent to $1\pmod n$.

Printed omission and usable range

The printed text of Lemma 3 reads: "The prime $P(n)$ can divide $\Phi_n$ to at most the first power. All other prime factors of $\Phi_n$ are congruent to $1 (\mathrm{mod}\, n)$." (p. 429). It omits an explicit $n>2$ qualifier. Taken literally at $n=2$, its first sentence is false: for $a=3$ and $b=1$, one has $P(2)=2$ and $\Phi_2(3,1)=3+1=4$, so $P(2)$ divides $\Phi_2(3,1)$ to the second power. Theorem 1 assumes $n>2$, and the Section 3 application begins with $n$ sufficiently large. This page therefore records the lemma only in the usable range $n>2$. This is a transparent compilation qualification, not an author-issued erratum.

Source and proof pointer

This is Lemma 3 on printed p. 429, the right half of physical p. 2 of the retained published scan. Stewart records it as a consequence of Birkhoff and Vandiver's work and credits a first version, apparently, to Sylvester; the paper gives no separate proof at this point. The lemma is used in Sections 3--4 for Theorems 1--2.

This page records the statement and source attribution only. Neither a proof of the lemma nor the cited predecessor arguments are transcribed or verified here.

Bears on. #977.