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Fix relatively prime integers $a>b>0$, write

$$\Phi_n(a,b)= \prod_{\substack{1\leq j\leq n\\(j,n)=1}}(a-\zeta^j b), \qquad P_n=P(\Phi_n(a,b)),$$

where $\zeta$ is a primitive $n$th root of unity and $P(m)$ is the greatest prime factor of $m$.

Statement

For every real $x$ with $0<x<1/\log 2$, there is a function $f$, strictly increasing and unbounded and explicitly specifiable in terms of $a,b,x$ only, such that

$$\frac{P_n}{n}>f(n) \tag{1}$$

for every integer $n>2$ having at most $x\log\log n$ distinct prime factors.

The paragraph following the theorem notes that almost all integers have $(1+o(1))\log\log n$ distinct prime factors. Choosing $1<x<1/\log 2$ therefore gives a covered set of natural density one that contains every sufficiently large prime. Since $a^n-b^n=\prod_{d\mid n}\Phi_d(a,b)$, the paper obtains $P(a^n-b^n)/n\to\infty$ along those exponents, including along the prime exponents.

Source and proof pointer

The statement is Theorem 1 on printed p. 427, the right half of physical p. 1 of the retained published scan. Its density-one and $a^n-b^n$ consequences are on printed p. 428, the left half of physical p. 2. The proof is Section 3 on printed pp. 429--431, running from the right half of physical p. 2 through the right half of physical p. 3.

The proof invokes the Baker estimate stated as Lemma 1 (Baker [2]) and the cyclotomic prime-divisor Lemma 3. Those arguments are not transcribed here; this page is a precise statement and proof pointer, not a complete-proof reconstruction.

Bears on. #977.