Solution: We seek integer solutions $ (x, y) $ to $ x^2 - y^2 = 2025 $.

["Solving for Integer Solutions: Finding $ (x, y) $ such that $ x^2 - y^2 = 2025 $", "The equation $ x^2 - y^2 = 2025 $ presents a classic Diophantine challenge, expressing the difference of two squares equal to a fixed positive integer. This form opens opportunities for factoring and generating integer solutions efficiently. In this article, we explore a clear, systematic solution method to determine all integer pairs $ (x, y) $ satisfying $ x^2 - y^2 = 2025 $.", "---", "### Understanding the Difference of Squares", "Recall the algebraic identity:", "$$\nx^2 - y^2 = (x + y)(x - y)\n$$", "Given $ x^2 - y^2 = 2025 $, we rewrite the equation as:", "$$\n(x + y)(x - y) = 2025\n$$", "Let $ a = x + y $ and $ b = x - y $. Then $ ab = 2025 $, and since $ x $ and $ y $ are integers, $ a $ and $ b $ must both be integers and of the same parity (both even or both odd) because:", "- $ x = \frac{a + b}{2} $\n- $ y = \frac{a - b}{2} $", "For $ x $ and $ y $ to be integers, $ a + b $ and $ a - b $ must be even, so $ a $ and $ b $ must have the same parity.", "---", "### Step 1: Factorizing 2025", "Begin by factoring 2025:", "$$\n2025 = 25 \ imes 81 = 5^2 \ imes 3^4\n$$", "So the prime factorization is $ 3^4 \cdot 5^2 $. The total number of positive divisors is:", "$$\n(4+1)(2+1) = 15\n$$", "Thus, 2025 has 15 positive divisors, and including negative divisors, there are 30 total integer divisors.", "We list all factor pairs $ (a, b) $ such that $ ab = 2025 $, including negative pairs:", "$$\n(a, b) = (\pm d, \pm \frac{2025}{d}) \quad \ ext{with } d \mid 2025\n$$", "We consider both signs, but only pairs where $ a $ and $ b $ have the same parity yield integer $ x, y $.", "---", "### Step 2: Parity Analysis", "Note that 2025 is odd, so all its divisors are odd. Therefore, both $ a $ and $ b $ are odd for every factor pair. Since odd + odd = even and even − even = even, both $ a + b $ and $ a - b $ are even — satisfying the parity condition.", "Hence, all factor pairs $ (a, b) $ of 2025 yield integer $ x, y $.", "---", "### Step 3: Expressing $ x $ and $ y $ in Terms of $ a $ and $ b $", "From $ a = x + y $, $ b = x - y $, solve:", "$$\nx = \frac{a + b}{2}, \quad y = \frac{a - b}{2}\n$$", "For each divisor $ d $ of 2025, set $ a = d $, $ b = \frac{2025}{d} $, and compute:", "$$\nx = \frac{d + \frac{2025}{d}}{2}, \quad y = \frac{d - \frac{2025}{d}}{2}\n$$", "But since $ d $ and $ \frac{2025}{d} $ are both odd, $ x $ and $ y $ are integers.", "---", "### Step 4: Generating All Integer Solutions", "Let’s list all divisors $ d $ of 2025 (positive and negative). The positive divisors are:", "$$\n1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025\n$$", "For each $ d $, $ a = d $, $ b = 2025/d $, and we compute:", "$$\nx = \frac{d + \frac{2025}{d}}{2}, \quad y = \frac{d - \frac{2025}{d}}{2}\n$$", "Because each factor pair $ (a, b) $ generates a solution $ (x, y) $, and $ (-a, -b) $ gives $ x \ o -x, y \ o -y $, we get symmetric solutions.", "Instead of computing each by hand, we can generalize:", "Each positive divisor $ d $ gives one solution where $ a = d $, $ b = 2025/d $, and one with $ a = -d $, $ b = -2025/d $, producing two distinct integer pairs unless $ x = y = 0 $ (not applicable here).", "Total of 15 positive divisors → 15 positive pairs $ (a,b) $, and 15 negative → total 30 ordered factor pairs.", "Each such pair gives a unique integer solution $ (x, y) $.", "But due to symmetry, many may repeat in magnitude but vary in sign.", "Let’s compute a few representative solutions:", "- $ d = 1 $:\n $ a = 1, b = 2025 $\n $ x = \frac{1 + 2025}{2} = 1013 $, $ y = \frac{1 - 2025}{2} = -1012 $", "- $ d = 3 $:\n $ a = 3, b = 675 $\n $ x = 339, y = -336 $", "- $ d = 5 $:\n $ a = 5, b = 405 $\n $ x = 205, y = -200 $", "- $ d = 9 $:\n $ a = 9, b = 225 $\n $ x = 117, y = -108 $", "- $ d = 15 $:\n $ a = 15, b = 135 $\n $ x = 75, y = -60 $", "- $ d = 25 $:\n $ a = 25, b = 81 $\n $ x = 53, y = -28 $", "- $ d = 27 $:\n $ a = 27, b = 75 $\n $ x = 51, y = -24 $", "- $ d = 45 $:\n $ a = 45, b = 45 $\n $ x = 45, y = 0 $", "- $ d = 75 $:\n $ a = 75, b = 27 $ → $ x = 51, y = 24 $", "- $ d = 81 $:\n $ a = 81, b = 25 $ → $ x = 53, y = 28 $", "- $ d = 135 $:\n $ a = 135, b = 15 $ → $ x = 75, y = 60 $", "- $ d = 225 $:\n $ a = 225, b = 9 $ → $ x = 117, y = 108 $", "- $ d = 405 $:\n $ a = 405, b = 5 $ → $ x = 205, y = 200 $", "- $ d = 675 $:\n $ a = 675, b = 3 $ → $ x = 339, y = 336 $", "- $ d = 2025 $:\n $ a = 2025, b = 1 $ → $ x = 1013, y = 1012 $", "For negative divisors:", "- $ d = -1 $: $ a = -1, b = -2025 $ → $ x = -1013, y = -1012 $\n- $ d = -3 $: $ x = -339, y = -336 $, etc.", "Thus, each divisor $ d $ gives a unique solution. Since 2025 has 15 positive and 15 negative divisors, we get 30 integer solutions in total.", "---", "### Final Answer: All Integer Solutions", "The complete set of integer solutions $ (x, y) $ to $ x^2 - y^2 = 2025 $ is given by:", "$$\nx = \frac{d + \frac{2025}{d}}{2}, \quad y = \frac{d - \frac{2025}{d}}{2}\n$$", "for each divisor $ d $ of 2025 (positive and negative). There are 30 such solutions.", "Explicitly, all solutions include:", "- $ (x, y) = (\pm1013, \pm1012) $, $ (\pm339, \pm336) $, $ (\pm205, \pm200) $, $ (\pm117, \pm108) $, $ (\pm75, \pm60) $, $ (\pm53, \pm28) $, $ (45, 0) $, $ (\pm51, \pm24) $, $ (\pm53, \pm28) $ — careful to list distinct combinations.", "More carefully, the 30 solutions consist of symmetric pairs due to sign flips of $ a $ and $ b $, but each divisor $ d $ gives one solution, so list them clearly:", "$$\n\boxed{\n\begin{aligned}\n&\left( \frac{2025 + 1}{2}, \frac{1 - 2025}{2} \right) = (1013, -1012), \quad \n\left( \frac{1 + 2025}{2}, \frac{2025 - 1}{2} \right) = (1013, 1012) \\n&\left( \frac{675 + 3}{2}, \frac{3 - 675}{2} "]