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Home » Discover the Secret Garden of Jamali: Where Every Flower Tells a Story! - Abbey Badges

Discover the Secret Garden of Jamali: Where Every Flower Tells a Story! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 x^2 = 144(k^2 + 1).

📌 So $x^2 = 144(k^2 + 1) = (12)^2(k^2 + 1)$, hence $x = 12\sqrt{k^2 + 1}$. For $x$ to be integer, $k^2 + 1$ must be a perfect square. Let $k^2 + 1 = m^2$, so:

📌 m^2 - k^2 = 1 \Rightarrow (m - k)(m + k) = 1.

📌 Integer solutions: $m = 1, k = 0$ or $m = -1, k = 0$. So only $k = 0$.

📌 Then $y = 5(0) = 0$, and $x^2 = 144(0 + 1) = 144 \Rightarrow x = \pm 12$.

📌 Thus, the only lattice points are $(\pm 12, 0)$.

📌 Check: $\frac{144}{144} - \frac{0}{25} = 1$, valid.

📌 No other $k$ gives $k^2 + 1$ square. Try small $y$ directly:

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