Exercise 7.8.10 (Least solutions of $x^2 - 61y^2 = \pm 1$)

Question

Calculator problem.} Find the least positive solution of $x^2 - 61 y^2 = -1$ and also of $x^2 - 61y^2 = 1$.

Richard Ganaye · Accepted Answer

To hell with the calculators with an eight-digit display!
\begin{proof} 
With the program given in Problem 9, we obtain
\begin{verbatim}
(29718, 3805, -1)
(1766319049, 226153980)
\end{verbatim}
So the least positive solution of $x^2 - 61 y^2 = -1$ is $(29718, 3805)$, and the least positive solution of $x^2 - 61 y^2 = 1$ is $(1766319049, 226153980))$:
\begin{align*}
29718^2 - 61 \cdot 3805 &= -1,\
1766319049^2 -61 \cdot 226153980^2 &= 1.
\end{align*}
\end{proof}

Exercise 7.8.10 (Least solutions of $x^2 - 61y^2 = \pm 1$)

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Exercise 7.8.10 (Least solutions of $x^2 - 61y^2 = \pm 1$)

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