A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers - Ana Paula Chaves; Pavel Trojovský (Mathematics, 2020)
The sequence of the k-generalized Fibonacci numbers <semantics> ( <msubsup> F n ( k ) </msubsup> ) n </semantics> is defined by the recurrence <semantics> <msubsup> F n ( k ) </msubsup> = <msubsup> ∑ j = 1 k </msubsup> <msubsup> F n − j ( k ) </msubsup> </semantics> beginning with the k terms <semantics> 0 , … , 0 , 1 </semantics> . In this paper, we shall solve the Diophantine equation <semantics> <msubsup> F n ( k ) </msubsup> = ( <msubsup> F m ( l ) </msubsup> ) 2 + 1 </semantics> , in positive integers <semantics> m , n , k </semantics> and l.