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Claim. C. L. Stewart, The greatest prime factor of $a^n-b^n$, Acta Arithmetica 26 (1974/75), no. 4, 427--433. Let $a>b>0$ be relatively prime integers, let $P(m)$ be the greatest prime factor of $m$, and fix $\kappa$ with $0<\kappa<1/\log2$. Theorem 1 (printed p. 427) gives a strictly increasing unbounded function $f$, effectively determined by $a$, $b$ and $\kappa$, with $P(\Phi_n(a,b))/n>f(n)$ for every integer $n>2$ having at most $\kappa\log\log n$ distinct prime factors, where $\Phi_n(a,b)$ is the homogeneous cyclotomic factor of $a^n-b^n$. Since $\Phi_n(a,b)$ divides $a^n-b^n$, the transfer after equation (4) (printed p. 428) gives $P(a^n-b^n)/n\to\infty$ as $n$ runs through the integers $n>2$ with at most $\kappa\log\log n$ distinct prime factors. With $a=2$ and $b=1$ this is the limit of Problem 977 along those $n$.

Covers. The limit $P(2^n-1)/n\to\infty$ along the integers $n>2$ with at most $\kappa\log\log n$ distinct prime factors, for each fixed $0<\kappa<1/\log2$. For $1<\kappa<1/\log2$ this set has natural density one and contains every sufficiently large prime, as Stewart notes on printed pp. 427--428. It does not give the limit along all $n$, which Stewart's 2013 theorem proves (its claim page).

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica 26 (1974/75), 427--433. The site's PROVED label credits the 2013 theorem, not this paper, so no curator acceptance is listed.