Diophantine equations for additive Pell numbers in Pell, Pell–Lucas, and Modified Pell numbers

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This paper investigates the Diophantine equations arising from ternary additive problems of Pell, Pell–Lucas, and Modified Pell numbers. Specifically, we characterize all integer solutions to the equation ${P_n} + {P_m} + {P_r} = {X_k}$, $X \in \left\{ {{P},{Q},{R}} \right\}$, where ${P_i}$, ${Q_i}$, and ${R_i}$ denote the $i$-th terms of the Pell, Pell–Lucas, and Modified Pell sequences, respectively. By leveraging recurrence relations, Binet's formulas, and Carmichael's Primitive Divisor Theorem, we provide the first complete classification of solutions to this ternary additive problem. Our results reveal several parametric and singular solutions. Furthermore, we reduce prior results to binary sums of the form ${P_n} + {P_m} = {X_k}$ as special instances of our framework.