Let \( (P_m)_{m\ge 0} \) be the sequence of
Pell numbers given by the linear recurrence; \( P_0=0 \), \( P_1=1 \), and \( P_{m+2} = 2P_{m+1}+ P_m \) for all \( m\ge 0 \). In the paper under review, the authors prove the following theorem, which is the main result in the paper.
Theorem 1. All nonnegative integer solutions \( (m_1, m_2, m_3, m_4, n) \) of the Diophantine equation
\[ N=P_{m_1}+P_{m_2}+P_{m_3}+P_{m_4} = d\left(\dfrac{10^{n}-1}{9}\right) \quad \text{with} \quad d\in \{1, 2, \ldots, 9\} \]
have
\[ N\in\{0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,77,88,99, 111, 222, 444, 888, 999\}. \]
Theorem 1 is a part of the series of papers that are a continuation of the result of
F. Luca [Math. Commun. 17, No. 1, 1–11 (2012;
Zbl 1305.11008)]. The proof of Theorem 1 follows from a clever combination of the well-known properties of Pell sequence, the theory of linear forms in complex and \(p\)-adic logarithms of algebraic numbers á la Baker, and the reduction techniques involving the theory of continued fractions. Computations are done with the help of
Maple.
