On the exponential Diophantine equation ${\left(P_n^{(k)}\right)}^x+{\left(P_r^{(k)}\right)}^x=P_m^{(k)}$

Fallou Faye, Bernadette Faye

The Pell numbers satisfy $P_n^2+P_{n+1}^2=P_{2n+1}$, and Bravo, Luca and Pomeo recently found all the cases in which the sum of the $x$-th powers of two consecutive $k$-generalized…