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
gcd (f(n)an+g(n), f1(n)bn+g1(n)) < exp(ne){\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. |