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On a problem of Ruderman. (English summary) Amer. Math. Monthly 118 (2011), no. 7, 644–650.1930-0972 This paper shows in a brief and elegant way that there are only finitely many pairs (m, n) of integers with m> n ≥ 0 such that 2 m − 2 n divides 3 m − 3 n. The main ingredient is a result by Bugeaud, Corvaja and Zannier [see P. Corvaja and U. M. Zannier, Compositio Math. 131 (2002), no. 3, 319–340; MR1905026 (2003e:11076)] implying that gcd(2 m−n − 1, 3 m−n − 1) ≪ε 2 ε(m−n). Some elementary number theory, based on the observation that 2 is a primitive root modulo 3 α for all α ≥ 1, then yields an absolute bound on max{m, n}. This bound is ineffective, though, since the subspace theorem was invoked in Bugeaud, Corvaja and Zannier’s work. Addressing this issue, the authors moreover show that an effective version of the abc-conjecture would give an effective bound on max{m, n}. Such an effective bound could eventually solve the original problem of Ruderman this article is referring to, namely, to show that whenever m, n are integers such that m> n ≥ 0 and 2 m − 2 n divides 3 m − 3 n, then 2 m − 2 n divides x m − x n for all positive integers x

Topics:
1. A. Baker, Logarithmic forms and the abc conjecture, in Number Theory, Diophantine, Computational

Year: 2013

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