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Let (Ln) n≥0 be the Lucas sequence given by L0 = 2, L1 = 1 and Ln+2 = Ln+1 + Ln for n ≥ 0. In this paper, we are interested in finding all powers of three which are sums of two Lucas numbers, i.e., we study the exponential Diophantine equation Ln + Lm = 3 a in nonnegative integers n, m, and a. The proof of our main theorem uses lower bounds for linear forms in logarithms, properties of continued fractions, and a version of the Baker-Davenport reduction method in Diophantine approximation.
For univariate polynomials with complex coefficients there are many estimates about the roots of polynomials. Moreover, the result corresponding to the ``continuity of the zeroes which respect to the coefficients'' is generally obtained as a corollary of Rouch'e's theorem and is rarely precise. Here we prove an explicit result for algebraically closed fields with an absolute value, in any characteristic
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