Let \(a = x - y\) and \(b = x + y\), so \(ab = 2024\), and \(x = \frac{a + b}{2}\), \(y = \frac{b - a}{2}\).
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- Question:** How many lattice points lie on the hyperbola \(x^2 - y^2 = 2024\) in the Cartesian plane?
- We seek integer solutions \((x, y)\) to the equation \(x^2 - y^2 = 2024\).
- This factors as:
- For \(x\) and \(y\) to be integers, \(a + b\) and \(b - a\) must both be even, so \(a\) and \(b\) must have the same parity.
- Now factor 2024:
- = 2^3 \cdot 11 \cdot 23
