Stewart 2013 divisors lucas lehmer

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lemma_4_3: Under the Lucas--Lehmer hypotheses, for every unramified prime ideal above a large prime p not dividing αβ, the order of (α/β)^n-1 at that ideal is less than p exp(-log p/(51.9 log log p)) log|α| log n.

theorem_1_1: Bounds the largest prime factor of a Lucas--Lehmer cyclotomic factor and, by the paper's direct integer specialization, proves the full Erdős limit for 2^n-1.

theorem_1_2: For fixed integers a>b>0 and every prime p not dividing ab beyond an effective threshold, the p-adic order of a^n-b^n is less than p exp(-log p/(52 log log p)) log a plus the p-adic order of n.

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Cameron L. Stewart, On divisors of Lucas and Lehmer numbers, Acta Mathematica 211 (2013), 291--314, DOI 10.1007/s11511-013-0105-y. For the arXiv version, the arXiv record names arXiv's non-exclusive distribution license (arXiv:1008.1274), every other right reserved. The published Acta PDF prints "© 2013 by Institut Mittag-Leffler. All rights reserved" on its first page, every other right reserved.

The two editions read for this card label the result differently:

• The arXiv:1008.1274v1 PDF is the 18-page manuscript dated 6 August 2010. It labels the main result Theorem 1 and equations (7)--(8).
• The published Acta PDF has 24 physical pages, printed pp. 291--314. It labels the result Theorem 1.1 and equations (1.7)--(1.8).

The main theorem gives an effective lower bound

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

for all sufficiently large $n$ under its Lucas--Lehmer hypotheses. The paper then states the direct integer specialization: for fixed integers $a>b>0$, the same lower bound holds for $P(a^n-b^n)$ for all sufficiently large $n$, with the threshold depending on the number of distinct prime factors of $ab$. Taking $a=2$ and $b=1$ proves the full limit $P(2^n-1)/n\to\infty$, rather than only a limsup statement.

The proof compares a lower bound for $|\Phi_n(\alpha,\beta)|$ with upper bounds for the prime-power contributions, using estimates for complex and $p$-adic linear forms in logarithms. The exact statement, edition mapping, direct specialization, and proof locations are recorded in Theorem 1.1. No complete proof is transcribed.

The paper's second main result, Theorem 1.2 (printed p. 295), bounds $\operatorname{ord}_p(a^n-b^n)$ above by $p\exp(-\log p/(52\log\log p))\log a+\operatorname{ord}_p n$ for primes $p\nmid ab$ beyond an effective threshold. The paper says it follows from a special case of Lemma 4.3 (printed p. 304), the $p$-adic estimate that yields a crucial step in the proof of Theorem 1.1.

Bears on. #977: equation (1.8) with $a=2$, $b=1$, recorded on the Theorem 1.1 page, gives $P(2^n-1)/n\to\infty$, the limit the problem asks about; Lemma 4.3 yields a crucial step of that proof, and Theorem 1.2 follows from a special case of Lemma 4.3.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.