2Question: A paleontologist discovers a fossilized bone fragment whose shape is modeled by the hyperbola $\frac{x^2}{144} - \frac{y^2}{25} = 1$. How many lattice points (points with integer coordinates) lie on this hyperbola?

["Understanding the Curve: The Hyperbola in Context\nBy a Fossil and Science Insight", "When a paleontologist uncovers a fossilized bone fragment whose shape closely follows a mathematical curve, it often opens a fascinating bridge between ancient life and modern geometry. In this case, the fossil’s structure is modeled by the hyperbola given by the equation:\n[\n\frac{x^2}{144} - \frac{y^2}{25} = 1\n]\nThis is a standard hyperbola opening horizontally, centered at the origin, with $a^2 = 144 \Rightarrow a = 12$ and $b^2 = 25 \Rightarrow b = 5$. Understanding the geometry of this curve helps us determine how many lattice points—integer coordinate pairs $(x, y)$—lie exactly on it.", "Lattice points on conic sections like hyperbolas are rare and precisely defined, making them ideal for mathematical exploration. To find such points, we substitute integer values for $x$ and solve for $y^2$, checking whether the result is a perfect square.", "Deriving the Condition for Integer Solutions\nStarting from the hyperbola equation:\n[\n\frac{x^2}{144} - \frac{y^2}{25} = 1\n]\nMultiply both sides by $144 \ imes 25 = 3600$ to eliminate denominators:\n[\n25x^2 - 144y^2 = 3600\n]\nRearranging gives:\n[\n25x^2 - 3600 = 144y^2\n]\nWe want $y^2$ to be a non-negative perfect square, so:\n[\ny^2 = \frac{25x^2 - 3600}{144}\n]\nFor $y$ to be an integer, the numerator $25x^2 - 3600$ must be divisible by 144 and result in a perfect square.", "Thus, $25x^2 - 3600 \equiv 0 \pmod{144}$.\nThis implies:\n[\n25x^2 \equiv 3600 \pmod{144}\n]\nSince $3600 \div 144 = 25$, we have $3600 \equiv 0 \pmod{144}$, so:\n[\n25x^2 \equiv 0 \pmod{144}\n]\nNow, $25$ and $144$ are coprime ($\gcd(25, 144) = 1$), so $x^2$ must be divisible by $144$. Thus, $x$ must be divisible by $12$, the least common multiple factor. Let:\n[\nx = 12k, \quad k \in \mathbb{Z}\n]\nSubstitute into the equation:\n[\n\frac{(12k)^2}{144} - \frac{y^2}{25} = 1 \Rightarrow \frac{144k^2}{144} - \frac{y^2}{25} = 1 \Rightarrow k^2 - \frac{y^2}{25} = 1\n]\nThen:\n[\n\frac{y^2}{25} = k^2 - 1 \Rightarrow y^2 = 25(k^2 - 1)\n]\nFor $y^2$ to be a perfect square, $25(k^2 - 1)$ must be a perfect square. Since 25 is already a perfect square ($5^2$), $k^2 - 1$ must also be a perfect square.", "Let $k^2 - 1 = m^2$, so:\n[\nk^2 - m^2 = 1 \Rightarrow (k - m)(k + m) = 1\n]\nThe only integer solutions to this equation are $k - m = 1$, $k + m = 1$ → adding gives $2k = 2 \Rightarrow k = 1$, $m = 0$\nOr $k - m = -1$, $k + m = -1$ → $k = -1$, $m = 0$", "Thus, the only integer values of $k$ that make $y^2$ a perfect square are $k = \pm1$", "For $k = \pm1$:\n- $x = 12k = \pm12$\n- $y^2 = 25(k^2 - 1) = 25(1 - 1) = 0 \Rightarrow y = 0$", "So the only lattice points are:\n[\n(12, 0) \quad \ ext{and} \quad (-12, 0)\n]", "Verifying No Other Lattice Points Exist\nWe check whether larger $|k|$ could yield perfect square $y^2$. Try small $k$ values:", "- $k = 0$: $y^2 = 25(-1)$ → invalid\n- $k = \pm2$: $y^2 = 25(4 - 1) = 75$ → not a perfect square\n- $k = \pm3$: $y^2 = 25(9 - 1) = 200$ → not a perfect square\n- $k = \pm4$: $y^2 = 25(16 - 1) = 375$ → no\n- $k = \pm5$: $25(25 - 1) = 600$ → no\n- As $|k|$ increases, $25(k^2 - 1)$ grows, but only perfect squares count—and no other such $k$ satisfy this identity due to the Diophantine nature of $k^2 - m^2 = 1$, which only has trivial solutions in integers.", "Thus, only two lattice points lie on the hyperbola.", "Conclusion: A Sparse Trail of Points\nWhile hyperbolas often have infinite or symmetrically rich structures, this particular fossil’s mathematical model restricts lattice points strictly to $(\pm12, 0)$. This elegant outcome reflects how geometric constraints tightly limit discrete point occurrences—offering paleontologists and mathematicians alike a precise, beautiful intersection of fossils and function.", "[\n\boxed{2}\n]"]