Abstract:
In this study, we find all Fibonacci numbers F-k and Lucas numbers L-k which are products of two Jacobsthal-Lucas numbers. More generally, taking k, m, n as nonnegative integers, we proved that F-k = j(m)j(n) = (2(m) + ((-1)(m)) . (2(n) + ((-1)(n)) implies that (k, m, n) = (1, 1, 1); (2, 1, 1), (3, 0, 1), (5, 1, 2), (9, 0, 4) and L-k = j(m)j(n) implies that (k, m, n) = (3, 0, 0), (0, 0, 1), (1, 1, 1), (4, 1, 3), As a result of this study, we showed that the largest Fibonacci number and Lucas number which can be written in the form (2(m) + ((-1)(m)) . (2n + (n) are F-9 = 34 = 2.17 = (2(0) + (-1)(0)) . (2(4) + ((-1)(4)) and L-4 = 7 = 1.7 = (2(1) + ((-1)(1)) . (2(3) + ((-1)(3)), respectively. Moreover the largest Fibonacci number and Lucas number which can be written in the form 2(n) + (-1)(n) are F-5 = 5 = 2(2) + (-1)(2) and L-4 = 7 = 2(3) + (-1)(3), respectively. As a result, it is shown that the only Fermat numbers in the Fibonacci sequence are F-3 = 3 and F-5 = 5 and the only Fermat number in the Lucas sequence is L-2 = 3. The proofs depend on lower bounds for linear forms and some tools from Diophantine approximation.
