Identities for products and squares of \((s,t)\)-Fibonacci and \((s,t)\)-Lucas numbers (Q2796816)

The Tribonacci sequence $\{ T_n \}_{n \geq 0}$ is given byNEWLINE\[CARRIAGE_RETURNNEWLINET_0 = 0, \ T_1 = T_2 = 1, \ T_{n+3} = T_{n+2} + T_{n+1} + T_nCARRIAGE_RETURNNEWLINE\]NEWLINEfor each natural number $n \geq 0$.NEWLINENEWLINEIn the paper, the following very interesting result is proved.NEWLINEFor each squarefree integer $d \not\in \{2, 3, 5, 15, 26\}$, the Pell equation $X^2 - dY^2 = \pm 1$ has at least two positive integer solutions $(X, Y)$ and $(X', Y')$ such that both $X$ and $X'$ are sums of two Tribonacci numbers.