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For a nonzero integer $x$ and a prime $p$, $\operatorname{ord}_p x$ is the exponent of $p$ in $x$, and $\omega(m)$ is the number of distinct prime factors of $m$.

Statement

Theorem 1.2 (printed p. 295). Let $a$ and $b$ be integers with $a>b>0$. There is a number $C_1$, effectively computable in terms of $\omega(ab)$, with the following property: for every prime $p$ that does not divide $ab$ and exceeds $C_1$, and every integer $n\ge2$,

$$\operatorname{ord}_p(a^n-b^n)< p\exp\!\left(-\frac{\log p}{52\log\log p}\right)\log a+\operatorname{ord}_p n. \tag{1.9}$$

Immediately after the theorem the paper records the case $n=p-1$: if $a>b>0$ are integers and $p$ is an odd prime not dividing $ab$ with $p>C_1$, then

$$\operatorname{ord}_p(a^{p-1}-b^{p-1})< p\exp\!\left(-\frac{\log p}{52\log\log p}\right)\log a .$$

The printed hypotheses of this consequence also include "$n$ is an integer with $n\geqslant 2$" (p. 295), although $n$ does not occur in its inequality.

The paper says the theorem follows from a special case of Lemma 4.3, the $p$-adic estimate that yields a crucial step in the proof of Theorem 1.1. It then cites Yamada's estimate $\operatorname{ord}_p(a^{p-1}-1)<C_2\,p(\log p)^{-2}\log a$, with $C_2$ effectively computable in terms of $\omega(a)$, display (1.10) on p. 296.

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 theorem and (1.9) are on printed p. 295 (physical p. 5), with the case $n=p-1$ below them. The proof is Section 6, printed p. 311 (physical p. 21).

In the arXiv:1008.1274v1 manuscript the result is Theorem 2 with display (9), physical p. 4. There the remark that follows also states the intermediate inequality $\operatorname{ord}_p(a^n-b^n)\le\operatorname{ord}_p(a^{p-1}-b^{p-1})+\operatorname{ord}_p n$ for odd $p\nmid ab$ and $n\ge2$, before the case $n=p-1$. The proof is Section 6, physical pp. 16--17.

In outline, the proof reduces to coprime $a,b$, shows that for an odd prime $p$ the $p$-adic order of $a^n-b^n$ is at most that of $a^{p-1}-b^{p-1}$ plus $\operatorname{ord}_p n$ (the paper's (6.5)), and then applies Lemma 4.3 with exponent $p-1$. 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. The proof was not checked line by line.

Bears on

• Problem 977: the theorem is not the result that settles the problem. It follows from a special case of Lemma 4.3, which yields a crucial step in the paper's proof of Theorem 1.1, whose specialization (1.8) with $a=2$, $b=1$ gives $P(2^n-1)/n\to\infty$.