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Statement

Setting (Section 4, printed p. 302): $\alpha$ and $\beta$ are complex numbers such that $(\alpha+\beta)^2$ and $\alpha\beta$ are nonzero integers and $\alpha/\beta$ is not a root of unity, labelled so that $|\alpha|\ge|\beta|$. For a positive integer $m$, $\omega(m)$ is the number of distinct prime factors of $m$.

Lemma 4.3 (printed p. 304). Let $n>1$ be an integer, let $p$ be a prime not dividing $\alpha\beta$, and let $\wp$ be a prime ideal of the ring of algebraic integers of $\mathbb Q(\alpha/\beta)$ that lies above $p$ and does not ramify. There is a positive number $C$, effectively computable in terms of $\omega(\alpha\beta)$ and the discriminant of $\mathbb Q(\alpha/\beta)$, such that whenever $p>C$,

$$\operatorname{ord}_{\wp}\!\left(\left(\frac{\alpha}{\beta}\right)^{n}-1\right)< p\exp\!\left(-\frac{\log p}{51.9\log\log p}\right)\log|\alpha|\log n .$$

The paper says this lemma yields a crucial step in the proof of Theorem 1.1 (p. 295). In Section 5 it gives display (5.4): for $n>c_2$ and each prime $p$ dividing $\Phi_n(\alpha,\beta)$ other than $P(n/(3,n))$, it bounds $\operatorname{ord}_{\wp}((\alpha/\beta)^n-1)$, which by (5.3) bounds $\operatorname{ord}_p\Phi_n(\alpha,\beta)$ (p. 310). Theorem 1.2 follows from (6.5) on taking $n=p-1$ in the lemma (p. 311).

Source and proof pointer

Cameron L. Stewart, On divisors of Lucas and Lehmer numbers, Acta Mathematica 211 (2013), 291--314, as identified on the source card. The lemma is stated on printed p. 304 (physical p. 14); its proof runs from p. 304 to p. 309 (physical pp. 14--19). In the arXiv:1008.1274v1 manuscript the same statement is Lemma 8, physical p. 10, with the proof on physical pp. 10--15.

In outline, the proof writes $\alpha/\beta$ as a root of unity times a $2^v$th power $\theta^{2^v}$ in $\mathbb Q(\alpha/\beta)$ with $v$ maximal, and then applies Yu's lower bound for linear forms in $p$-adic logarithms (Section 3) to a linear form in which, as the paper itself remarks on p. 295, the number of terms is deliberately enlarged, to $k=\lfloor\log p/(51.8\log\log p)\rfloor$ terms (display (4.9)) by bringing in small auxiliary primes. The proof is not transcribed here.

Read depth. Claims checked: the statement, its hypotheses, constants, label and page were read clause by clause on the printed page, together with the standing assumptions of Section 4. The proof was not checked line by line.

Bears on

• Problem 977: the lemma supplies the bound (5.4) in the proof of Theorem 1.1, whose integer specialization (1.8) with $a=2$, $b=1$ gives $P(2^n-1)/n\to\infty$. The paper notes (p. 296) that Yamada's estimate (1.10) can take the lemma's place and gives the weaker bound (1.11), which also yields the limit.