In this study, we investigated a Fermat-like Diophantine equation involving Pell numbers, specifically examining powers of two that can be represented as the sum of the $ x $-th powers of any two Pell numbers. To solve the equation $ {P^{x}_m} + {P^{x}_n} = {2^a} $ for $ x \geq 3 $, where $ m $, $ n $, $ x $, and $ a $ are non-negative integers, we employed Baker's theory of linear forms in logarithms, a modified version of the Baker-Davenport reduction method, and properties of continued fractions. Our results extend previous findings for $ x = 1 $ and $ x = 2 $, shedding light on the interplay between Pell numbers and powers of two in this exponential context.
Citation: Ahmet Emin. On the dynamics of the Fermat-like Diophantine equation involving Pell numbers[J]. AIMS Mathematics, 2025, 10(8): 18524-18540. doi: 10.3934/math.2025827
In this study, we investigated a Fermat-like Diophantine equation involving Pell numbers, specifically examining powers of two that can be represented as the sum of the $ x $-th powers of any two Pell numbers. To solve the equation $ {P^{x}_m} + {P^{x}_n} = {2^a} $ for $ x \geq 3 $, where $ m $, $ n $, $ x $, and $ a $ are non-negative integers, we employed Baker's theory of linear forms in logarithms, a modified version of the Baker-Davenport reduction method, and properties of continued fractions. Our results extend previous findings for $ x = 1 $ and $ x = 2 $, shedding light on the interplay between Pell numbers and powers of two in this exponential context.
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Ahmet Emin. On the dynamics of the Fermat-like Diophantine equation involving Pell numbers[J]. AIMS Mathematics, 2025, 10(8): 18524-18540. doi: 10.3934/math.2025827
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