JP Journal of Algebra, Number Theory and Applications
Volume 14, Issue 2, Pages 157 - 175
(August 2009)
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SOLUTIONS OF SOME NONLINEAR DIOPHANTINE MATRIX EQUATIONS
Claude Gauthier (Canada) and Gérard Kientega (Burkina Faso)
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Abstract: Matrix generalizations of some classical nonlinear Diophantine equations of the theory of numbers are examined. We show that when the general solution or a formula for a class of solutions of the regular Diophantine equation is known, then its matrix generalization is easy to solve by simply replacing the parameters used to express these solutions by in pair commutative square matrices. Here, this method is applied to the Pythagorean, Aubry’s and Euler’s matrix equations. But the solutions obtained in this way are not the most general possible. We obtain the general solutions of the Pythagorean equation and Fermat’s equation of degrees 3 and 4 in the set of square matrices of order 2. These solutions show that the matrix generalizations of Fermat-Wiles’ theorem, and thus of Beal’s conjecture, are false. Many open problems are pointed out. |
Keywords and phrases: Diophantine matrix equations, Fermat’s matrix equation, Beal’s conjecture. |
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