On Mixed 𝐵-Concatenations of Pell and Pell–Lucas Numbers which are Pell Numbers

Let (𝑃 𝑛 ) 𝑛≥0 and (𝑄 𝑛 ) 𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃 𝑛 = 𝑏 𝑑 𝑃 𝑚 + 𝑄 𝑘 and 𝑃 𝑛 = 𝑏 𝑑 𝑄 𝑚 + 𝑃 𝑘 with 𝑑, the number of digits of 𝑃 𝑘 or 𝑄 𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.