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Shifting Lucas Sequences Away from Primes
Summary: We strengthen a result of Jones by showing that for any positive integer \(P\), the Lucas sequence \((U_n)_n\) defined by \(U_0 = 0\), \(U_1 = 1\), \(U_n=P \cdot U_{n -1} + U_{n - 2}\) can be translated by a positive integer \(K(P)\) such that the shifted sequence with general term \(U_n + K(P)\) contains no primes, nor terms one unit away ...Ismailescu, Dan +4 more
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Mersenne Numbers in Generalized Lucas Sequences
Proceedings of the Bulgarian Academy of SciencesLet $$k \geq 2$$ 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.
Altassan, Alaa, ALAN, Murat
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Lucas Sequences in Primality Testing
2014Prime or composite? This classification determines whether or not integers can be used in digital security. One such way to begin testing an integers primality is with the Fermat test, which says that if n is a prime number and a is an integer then an-1 1 mod n.
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Some Generalized Lucas Sequences
The Fibonacci Quarterly, 1985J. H. Clarke, A. G. Shannon
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