Generalized Fibonacci numbers of the form \(wx^2 + 1\).(English)Zbl 1399.11044
Summary: Let \(P \geq 3\) be an integer and let \((U_n)\) and \((V_n)\) denote generalized Fibonacci and Lucas sequences defined by \(U_0 = 0, U_1 = 1; V_0 = 2, V_1 = P\), and \(U_{n+1} = PU_n - U_{n-1}\), \(V_{n+1} = PV_n - V_{n-1}\) for \(n \geq 1\). In this study, when \(P\) is odd, we solve the equation \(U_n = wx^2 +1\) for \(w = 1, 2, 3, 5, 6, 7, 10\). After then, we solve some Diophantine equations utilizing solutions of these equations.
MSC:
| 11B37 | Recurrences |
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11B50 | Sequences (mod \(m\)) |
| 11B99 | Sequences and sets |
| 11D41 | Higher degree equations; Fermat’s equation |
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