Abstract.
In [2], it was shown that if a and b are multiplicatively independent integers and ɛ > 0, then the inequality gcd (an − 1,bn − 1) < exp(ɛn) holds for all but finitely many positive integers n. Here, we generalize the above result. In particular, we show that if f(x),f1(x),g(x),g1(x) are non-zero polynomials with integer coefficients, then for every ɛ > 0, the inequality \({\rm gcd}\, (f(n)a^n+g(n), f_1(n)b^n+g_1(n)) < \exp(n\varepsilon)\) holds for all but finitely many positive integers n.
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Luca, F. On the Greatest Common Divisor of u − 1 and v − 1 with u and v Near \({\cal S}\)-units. Mh Math 146, 239–256 (2005). https://doi.org/10.1007/s00605-005-0303-6
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DOI: https://doi.org/10.1007/s00605-005-0303-6