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A generalization of the well–known Fibonacci sequence is the ℓ -Fibonacci sequence (F m (ℓ)) m whose first ℓ terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding ℓ terms. The k -Pell sequence (P n (k)) n , which is a generalization of the classical Pell sequence, can be defined similarly. In this paper, we find all coincidences between these two families of sequences. That is, we find all the solutions of the Diophantine equation P n (k) = F m (ℓ) in positive integers n , k , m , ℓ with k , ℓ ≥ 2. This paper continues and extends prior results which dealt with the above problem for some particular cases of k and ℓ. In particular, it extends the previous work [2] that found all Fibonacci numbers in the Pell sequence. [ABSTRACT FROM AUTHOR]

In this paper, we find all solutions to the Diophantine equation Fn + Fm = 2a(Fr + Fs), where {Fk}k=0 is the Fibonacci sequence. This paper continues and extends previous work, which investigated the powers of 2 that are sums of two Fibonacci numbers. [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), whereFkis thekth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation. [ABSTRACT FROM PUBLISHER]

In 2006, F. Luca and I. E. Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) proved that there are only finitely many pairs (n, m) of positive integers which satisfy the Diophantine equation |τ(n!)|=m!, where τ is the Ramanujan function. In this paper, we follow the same approach of Luca and Shparlinski (Proc. Indian Acad. Sci. (Math. Sci.)116(1) (2006), 1–8) to determine all solutions of the above equation. The proof of our main theorem uses linear forms in two logarithms and arithmetic properties of the Ramanujan function. [ABSTRACT FROM PUBLISHER]

Abstract: The k-generalized Fibonacci sequence resembles the Fibonacci sequence in that it starts with (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we are interested in finding the integers that appear in different generalized Fibonacci sequences; i.e., we study the Diophantine equation in positive integers n, k, m, ℓ with . [Copyright &y& Elsevier]