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For an integer $m$, let $P(m)$ be its greatest prime factor, with the source's convention $P(m)=1$ for $m\in\{-1,0,1\}$, and let $\omega(m)$ be the number of distinct prime factors of $m$. Let $\Phi_n(\alpha,\beta)$ denote the homogeneous $n$th cyclotomic polynomial evaluated at $\alpha,\beta$.

Statement

Suppose that $\alpha,\beta\in\mathbb C$, that $(\alpha+\beta)^2$ and $\alpha\beta$ are nonzero integers, and that $\alpha/\beta$ is not a root of unity. Then some positive constant $C$, which can be computed effectively from $\omega(\alpha\beta)$ and the discriminant of the field $\mathbb Q(\alpha/\beta)$, has the property that every integer $n>C$ satisfies

$$P(\Phi_n(\alpha,\beta))> n\exp\!\left(\frac{\log n}{104\log\log n}\right). \tag{1}$$

The published paper immediately gives the following direct integer specialization. If $a>b>0$ are fixed integers, then

$$P(a^n-b^n)> n\exp\!\left(\frac{\log n}{104\log\log n}\right) \tag{2}$$

for all sufficiently large $n$, with the threshold depending on the number of distinct prime factors of $ab$.

For $a=2$ and $b=1$, (2) yields

$$\frac{P(2^n-1)}{n}> \exp\!\left(\frac{\log n}{104\log\log n}\right)\longrightarrow\infty.$$

Thus the specialization proves the full limit in Problem 977. This transfer uses the paper's direct equation (1.8); no additional cyclotomic-divisibility argument is needed.

Source and proof pointer

In the published Acta PDF, the theorem is Theorem 1.1 and (1) is equation (1.7), physical p. 4 / printed p. 294. The integer specialization (2) is equation (1.8) at the bottom of that page, with its threshold clause continuing on physical p. 5 / printed p. 295. The proof is Section 5, physical pp. 19--20 / printed pp. 309--310.

In the arXiv v1 manuscript, the same result is Theorem 1 and equation (7), physical p. 3; the integer specialization is equation (8), physical p. 4; and the proof is Section 5, physical pp. 15--16. These arXiv locators are not published-page locators.

Only the theorem statement, the source's stated specialization, and the elementary substitution $a=2,b=1$ are recorded here. Stewart's proof and its same-paper lemmas are not transcribed, so this page carries no complete-proof or proof-verification claim.

Bears on. #977: the specialization (2) with $a=2$, $b=1$ gives $P(2^n-1)/n\to\infty$, the limit the problem asks about. The proof of (1) uses Lemma 4.3 to bound $\operatorname{ord}_p\Phi_n(\alpha,\beta)$ for each prime $p$ dividing $\Phi_n(\alpha,\beta)$ other than $P(n/(3,n))$.