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Home » R = \frac4(\sqrtx - 2)(\sqrtx + 2)(\sqrtx - 2) - Dachbleche24

R = \frac4(\sqrtx - 2)(\sqrtx + 2)(\sqrtx - 2) - Dachbleche24

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 So $ \gcd = 4 \cdot 9 = 36 $

📌 $$Question: How many lattice points lie on the hyperbola defined by $ x^2 - y^2 = 2025 $?

📌 Solution: We are tasked with finding the number of integer solutions $(x, y)$ to the equation $ x^2 - y^2 = 2025 $. This can be factored as:

📌 Let $ a = x - y $ and $ b = x + y $. Then $ ab = 2025 $, and $ x = \frac{a + b}{2} $, $ y = \frac{b - a}{2} $. For $ x $ and $ y $ to be integers, $ a + b $ and $ b - a $ must both be even, which implies $ a $ and $ b $ must be both even or both odd.

📌 Now, factor 2025:

📌 = 3^4 \cdot 5^2

📌 The number of positive divisors is $ (4+1)(2+1) = 15 $, so there are 15 positive divisor pairs $ (a, b) $ with $ a > 0, b > 0 $, and another 15 with $ a < 0, b < 0 $, giving 30 total integer divisor pairs (since $ a $ and $ b $ can both be negative).

📌 But we only count pairs where $ a $ and $ b $ have the same parity. Since 2025 is odd, all its divisors are odd. Therefore, $ a $ and $ b $ are both odd in every factor pair, and the sum and difference are even. So all 30 pairs yield integer $ (x, y) $.

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