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Fix relatively prime integers $a>b>0$ and put $P_n=P(\Phi_n(a,b))$, where $P(m)$ denotes the greatest prime factor of $m$.

Statement

For every prime $p$ larger than a constant $C=C(a,b)$, which can be computed effectively from $a$ and $b$ alone,

$$P_p>\frac12p(\log p)^{1/4}, \qquad P_{2p}>p(\log p)^{1/4}.$$

Source and proof pointer

The statement is Theorem 2 on printed p. 428, the left half of physical p. 2 of the retained published scan. The proof is Section 4, beginning on printed p. 431 (physical p. 3, right half) and ending on printed p. 432 (physical p. 4, left half).

The proof uses the earlier Baker estimate stated as Lemma 2 and the cyclotomic prime-divisor Lemma 3. It is not transcribed here, so this page carries no complete-proof or proof-verification claim.

Bears on. #977.