2022
DOI: 10.48550/arxiv.2208.03068
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On the Diophantine equation $U_n-b^m = c$

Abstract: Let (Un) n∈N be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B and N 0 such that for any b, c ∈ Z with b > B the equation Un − b m = c has at most two distinct solutions (n, m) ∈ N 2 with n ≥ N 0 and m ≥ 1. Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1, T 3 = 2 and Tn = T n−1 + T n−2 + T n−3 for n ≥ 4. By means of the LLL-algorithm and continued fractio… Show more

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