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Diego Marques was awarded a PhD degree at the Universidade de Brasília (2009) in less than four months. Currently, he is a professor in the department of mathematics at the same university. His field of interest is elementary and transcendental number theory.Ana Paula Chaves received her B.S. and M.S. degrees in mathematics from the Universidade Federal do Ceará, and her PhD from Universidade de Brasília in 2013. Currently, she is a professor of mathematics and statistics at the Universidade Federal de Goiás. Her interests are transcendental number theory and diophantine equations.

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The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l.