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A Method of Solving a Dophantine Equation of Second Degree with N Variables

Florentin Smarandache
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Abstract

First, we consider the equation $ax^2 - by^2 + c = 0$, with $a,b \in N*$ and $c \in Z*$, which is a generalization of Pell's equation. Here, we show that: if this equation has an integer solution and $ab$ is not a perfect square, then it has infinitely many integer solutions; in this case we find a closed expression for $(x_{n}, y_{n})$, the general positive integer solution, by an original method. More, we generalize it for a Diophantine equation of second degree and with n variables of the form: $\sum_{i=1}^{n} a_{i}x_{i}^{2} = b$.

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Florentin Smarandache. 2004-05-12. A Method of Solving a Dophantine Equation of Second Degree with N Variables. https://arxiv.org/abs/math/0405206

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