The 2-adic valuation of generalized Fibonacci sequences with an application to certain Diophantine equations

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In this paper we focus on finding all the factorials expressible as a product of a fixed number of 2k-nacci numbers with k ≥ 2. We derive the 2-adic valuation of the 2k-nacci sequence and use it to establish bounds on the solutions of the initial equation. In addition, we specify a more general family of sequences, for which we can perform a similar procedure. We also investigate a possible connection of these results with p-regular sequences.