Schinzel nd two theorems gelfond applications

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Schinzel, A., On two theorems of Gelfond and some of their applications. Acta Arith. 13 (1967/68), 177-236. 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-13-2-177-236, read 2026-10-02): the Creative Commons Attribution license with no version named.

Schinzel reworks Gelfond's ordinary and p-adic measures of irrationality for the ratio of two logarithms of algebraic numbers so that the resulting bounds are explicit rather than depending on unspecified functions n_0(eps, alpha, beta). Section 2 reproduces Gelfond's 1940 arguments with modifications replacing log^{3+eps} max{|n|,|m|} by C(alpha,beta)(log max{|n|,|m|} + C'(alpha,beta))^3, or its p-adic analogue, in the inequalities for G_0 and G_p; Theorems 1 (p-adic) and 2 (ordinary) write these constants out explicitly, and when log|alpha|/log|beta| is rational he obtains G_0 > -C(alpha,beta)(log max{|n|,|m|} + C'(alpha,beta))^2, the paper's one improvement on Gelfond's 1952 work, reformulated as Corollary 1 in Diophantine-approximation terms. Section 3 applies this to second-order linear recurrences: Theorems 3 and 4 give explicit lower estimates for |u_n| when the companion polynomial has non-real roots, comprising earlier results of P. Chowla, S. Chowla, Dunton, Lewis, Townes and Schinzel. Theorem 5 shows that for a negative odd integer d other than 1 - 2^k the equation x^2 - d = 2^m has at most one solution with m > 80 and x > 0, so that the Browkin-Schinzel conjecture (at most one solution in positive integers for d other than 1 - 2^k and -23) reduces to a finite computation; Theorem 6 proves the analogue for x^2 - d = p^m with p a prime factor of 1 - 4d, and Theorems 7 and 8 estimate the greatest prime factor of u_n. Section 4 treats x^nu - eps P_1^{n_1} ... P_k^{n_k} with nu = 2 or 3, eps = ±1 and P_i positive integers: Theorem 9 bounds its absolute value from below, Theorem 10 bounds its greatest prime factor from below for k <= 3 under stated restrictions, and Corollary 6 uses Theorem 10 to solve effectively Diophantine equations of the form q_1^{y_1} ... q_i^{y_i} ± r_1^{z_1} ... r_j^{z_j} = s^x, with the q's and r's distinct primes and s >= 1, where 6 does not divide s if the sign is minus. Corollary 5 to Theorem 9 gives, for a real quadratic irrational xi and an integer g > 1, the effective bound ||xi g^n|| > g^{-n} exp(c n^{1/7}), slightly more than Liouville's theorem yields. Section 5, "The greatest prime factor of a quadratic or cubic polynomial", bounds q(Ax^nu - E) from below by a multiple of log log x for nu = 2, 3 (Theorem 11, with Corollary 7 for any quadratic polynomial without a double root), improves the constants of Mahler and Nagell for Ax^2 - E with E | 4 (Theorem 12), and in Theorems 13-15 goes the opposite way, showing that for suitable x the greatest prime factor of Ax^nu - E, and more weakly of any integer polynomial f(x) of degree greater than 1, is small compared with the value; this is the aspect bearing on problems #368 and #928, which cite it as [Sc67b]. The paper closes with an open problem, and Schinzel notes that Baker's 1966 solution of the three-logarithms problem would generalize many of these results.

Source: https://doi.org/10.4064/aa-13-2-177-236.

Bears on. #368, #928

Results to transcribe.

• Theorems 1 and 2: Explicit versions of Gelfond's p-adic (Theorem 1) and ordinary (Theorem 2) irrationality measures for the ratio of two logarithms, with log^{3+eps} N, N = max{|n|,|m|}, replaced by an explicit constant times (log N + C')^3; in Theorem 2 the exponent 3 becomes 2 when |alpha| and |beta| are multiplicatively dependent (when alpha and beta are, if |alpha| = |beta| = 1).
• Corollary 1: For an algebraic integer gamma != 0 with gamma/|gamma| not a root of unity and H = max{|n_1|,|n_2|} > 1, |arg(gamma)/(2 pi) - n_1/n_2| > exp(-c(gamma) log^2 H), with c(gamma) independent of n_1, n_2.
• Theorems 3 and 4: Explicit lower estimates for |u_n| for second-order linear recurrences with non-real companion-polynomial roots, comprising earlier estimates of Chowla, Dunton, Lewis, Townes and Schinzel.
• Theorem 5: For a negative odd integer d != 1 - 2^k, the equation x^2 - d = 2^m has at most one solution with m > 80, x > 0.
• Theorem 10: Under each of four stated conditions, all with k <= 3, a lower bound for the greatest prime factor of x^nu - eps P_1^{n_1} ... P_k^{n_k}; Corollary 6 derives from it the effective solution of Diophantine equations q_1^{y_1} ... q_i^{y_i} ± r_1^{z_1} ... r_j^{z_j} = s^x (q's and r's distinct primes, s not divisible by 6 when the sign is minus).
• Theorems 11 and 12, Corollary 7 and Theorems 13-15: Lower bounds of order log log x for the greatest prime factor of quadratic and cubic binomials Ax^nu - E (Theorem 11) and of quadratic polynomials (Corollary 7), Theorem 12 improving Mahler's and Nagell's constants for Ax^2 - E with E | 4, together with upper bounds showing how small the greatest prime factor of a binomial (Theorems 13 and 14) or of a general integer polynomial (Theorem 15) can be for suitable x.