Solve \(x^2-5y^2=1\) by finding the fundamental solution from the continued fraction expansion of \(\sqrt{5}\), and check that \((9,4)\) satisfies the equation.
Choose one option.
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D. \((9,4)\) is the fundamental solution, because its \(y\) is the smallest positive
With \(\sqrt{5}=[2;\overline{4}]\), the first convergent satisfying the equation is \(9/4\), and \(9^2-5\cdot 4^2=81-80=1\). The option claiming \(4-5=1\) is a false computation, and the others do not satisfy the equation.
Pell's equation is the quadratic Diophantine equation in integers \(x\), \(y\) that sets equal to 1 the difference between \(x^2\) and \(d\) times \(y^2\), where \(d\) is a positive integer that is not a perfect square.