Stewart nd greatest prime factor

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lemma_3: For n>2, restricts the multiplicity and residue class of prime divisors of Phi_n(a,b).

theorem_1: Makes P(Phi_n(a,b))/n tend to infinity on a density-one family of exponents containing the primes.

theorem_2: Gives effective lower bounds for the largest prime factors of the pth and 2p-th homogeneous cyclotomic factors.

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C. L. Stewart, The greatest prime factor of $a^n-b^n$, Acta Arithmetica 26 (1974/75), no. 4, 427--433, DOI 10.4064/aa-26-4-427-433.

The retained published scan has four physical landscape images. Physical p. 1 contains printed p. 427 on its right; physical p. 2 contains printed pp. 428--429; physical p. 3 contains printed pp. 430--431; and physical p. 4 contains printed pp. 432--433. The scan identifies the volume as 1975, while the journal citation uses 1974/75. No notice is printed in the file; the publisher's article record offers the PDF "Free download under CC-BY license" ("Pobierz zgodnie z CC-BY" as the Polish page prints it), no version named (https://www.impan.pl/get/doi/10.4064/aa-26-4-427-433, read 2026-10-02): the Creative Commons Attribution license with no version named.

For relatively prime integers $a>b>0$, Stewart writes $P_n=P(\Phi_n(a,b))$. Theorem 1 proves that, for $0<x<1/\log 2$, one has $P_n/n>f(n)$ whenever $n>2$ has at most $x\log\log n$ distinct prime factors, where $f$ is strictly increasing, unbounded, and effectively specifiable from $a,b,x$. The paper states the density-one and prime-exponent conclusions. The compiler makes the choice $1<x<1/\log 2$ explicit: these exponents form a density-one set containing every sufficiently large prime; hence $P(a^n-b^n)/n\to\infty$ along that set. Theorem 2 gives the explicit prime and twice-prime estimates

$$P_p>\tfrac12p(\log p)^{1/4},\qquad P_{2p}>p(\log p)^{1/4}$$

for every sufficiently large prime $p$, with an effective threshold depending only on $a,b$. The proofs use Baker's estimates for linear forms in logarithms, the homogeneous cyclotomic factorization, and the prime-divisor structure stated as Lemma 3.

Bears on. #977.

Results.

• Theorem 1: few-prime-factor exponents
• Theorem 2: prime and twice-prime exponents
• Lemma 3: prime divisors of the cyclotomic factor

The result pages give statement and proof locators only; no proof from this paper is transcribed.