Generalized Fibonacci numbers of the form \(11x^2 + 1\).(English)Zbl 1430.11025
Summary: Let \(P \geq 3\) be an integer and let \((U_n)\) denote generalized Fibonacci sequence defined by \(U_0=0, U_1=1\) and \(U_{n+1}=P U_n - U_{n-1}\) for \(n\geq1\). In this study, when \(P\) is odd, we solve the equation \(U_n=11x^2+1\). We show that only \(U_1\) and \(U_2\) may be of the form \(11x^2+1\).
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
| 11A07 | Congruences; primitive roots; residue systems |
| 11D09 | Quadratic and bilinear Diophantine equations |
Cite
Full Text:DOI
References:
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