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MATRIX SOLUTIONS FOR THE NON-LINEAR EXPONENTIAL DIOPHANTINE EQUATION$$\left(X^a+\alpha I_q\right)^m+\left(Y^b+\beta I_q\right)^n=Z^2$$$$\alpha, \beta \in Z, a, b, m, n, q \in \mathrm{~N}, X, Y, Z \in M_q(\mathrm{~N})$$
Keywords:
Kolo F. Soro, Eric D. Akéké and Jean R. Tsiba, Matrix solutions for the non-linear exponential Diophantine equationDOI:
https://doi.org/10.17654/0972555526001Abstract
We investigate matrix solutions for the non-linear exponential Diophantine equation
$$
\left(X^a+\alpha I_q\right)^m+\left(Y^b+\beta I_q\right)^n=Z^2,
$$
where $\alpha, \beta \in \mathrm{Z}$ and $a, b, m, n, q \in \mathrm{~N}$ such that $q$ is a common multiple of $a$ and $b$. We show that this equation admits an infinite number of matrix solutions which do not depend on $m$ and $n$.
Received: April 25, 2025
Revised: July 12, 2025
Accepted: September 1, 2025
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