Mersenne Numbers in Generalized Lucas Sequences
DOI:
https://doi.org/10.7546/CRABS.2024.01.01Keywords:
$$k$$-generalized Lucas sequences, $$k$$-Lucas numbers, Mersenne numbers, linear forms in logarithms, Lucas numbersAbstract
Let $$k \geq 2$$ be an integer and let $$(L_{n}^{(k)})_{n \geq 2-k}$$ be the $$k$$-generalized Lucas sequence with certain initial $$k$$ terms and each term afterward is the sum of the $$k$$ preceding terms. Mersenne numbers are the numbers of the form $$2^a-1$$, where $$a$$ is any positive integer. The aim of this paper is to determine all Mersenne numbers which lie inside $$k$$-Lucas sequences.
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