Solution:** We seek integer solutions \( (x, y) \) to \( \frac{x^2}{25} + \frac{y^2}{9} = 1 \), or equivalently:

["# Integer Solutions to ( \frac{x^2}{25} + \frac{y^2}{9} = 1 ): Finding All ( (x, y) ) That Satisfy This Elliptic Equation", "## Understanding the Equation", "We are given the equation:", "[\n\frac{x^2}{25} + \frac{y^2}{9} = 1\n]", "This equation resembles the standard form of an ellipse:\n[\n\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\n]\nwith ( a = 5 ) and ( b = 3 ). While this looks like a smooth geometric shape, we are specifically interested in integer solutions ( (x, y) ), meaning both ( x ) and ( y ) are whole numbers.", "This type of Diophantine equation—involving integer solutions—has deep roots in number theory and geometry.", "---", "## Reécriture as a Diophantine Equation", "To find integer points ( (x, y) ) on this ellipse, we eliminate fractions by multiplying both sides by ( 25 \ imes 9 = 225 ):", "[\n9x^2 + 25y^2 = 225\n]", "Now we seek integer solutions ( (x, y) ) satisfying:", "[\n9x^2 + 25y^2 = 225\n]", "Since ( x^2 \geq 0 ) and ( y^2 \geq 0 ), we can bound the possible values of ( x ) and ( y ).", "---", "## Bounding the Variables", "From ( 9x^2 \leq 225 ), we get:", "[\nx^2 \leq \frac{225}{9} = 25 \quad \Rightarrow \quad |x| \leq 5\n]", "From ( 25y^2 \leq 225 ), we get:", "[\ny^2 \leq \frac{225}{25} = 9 \quad \Rightarrow \quad |y| \leq 3\n]", "So the possible integer candidates are:", "- ( x \in {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5} )\n- ( y \in {-3, -2, -1, 0, 1, 2, 3} )", "We now systematically check combinations satisfying ( 9x^2 + 25y^2 = 225 ).", "---", "## Systematically Searching Integer Solutions", "Let’s compute ( 9x^2 ) and ( 25y^2 ) for each ( y ), then check if the remainder is divisible by 9 and yields a perfect square.", "### Try ( y = 0 ):", "[\n25(0)^2 = 0 \Rightarrow 9x^2 = 225 \Rightarrow x^2 = 25 \Rightarrow x = \pm 5\n]\n✅ Solutions: ( (±5, 0) )", "### Try ( y = ±1 ):", "[\n25(1)^2 = 25 \Rightarrow 9x^2 = 200 \Rightarrow x^2 = \frac{200}{9} <br/>\not\in \mathbb{Z}\n]\n❌ No solution", "### Try ( y = ±2 ):", "[\n25(2)^2 = 100 \Rightarrow 9x^2 = 125 \Rightarrow x^2 = \frac{125}{9} <br/>\not\in \mathbb{Z}\n]\n❌ No solution", "### Try ( y = ±3 ):", "[\n25(3)^2 = 225 \Rightarrow 9x^2 = 0 \Rightarrow x^2 = 0 \Rightarrow x = 0\n]\n✅ Solutions: ( (0, ±3) )", "---", "## Final List of Integer Solutions", "From above, the only integer solutions are:", "[\n(x, y) = (5, 0),\ (-5, 0),\ (0, 3),\ (0, -3)\n]", "These are the only lattice points lying exactly on the ellipse defined by ( \frac{x^2}{25} + \frac{y^2}{9} = 1 ).", "---", "## Why These Are the Only Solutions", "The equation ( 9x^2 + 25y^2 = 225 ) places strict arithmetic constraints. Since ( x^2 ) and ( y^2 ) must be integers, and coefficients 9 and 25 are relatively large compared to the total sum, very few combinations work. Our exhaustive check confirms only four integer pairs satisfy the condition.", "While no closed-form formula characterizes all rational points on ellipses (unlike conics in projective geometry), for small coefficients and integer constraints, brute-force over bounded ranges remains effective.", "---", "## Geometric and Number-Theoretic Insight", "The ellipse ( \frac{x^2}{25} + \frac{y^2}{9} = 1 ) has semi-major axis 5 along the ( x )-axis, semi-minor axis 3 along the ( y )-axis. Integer points must lie on the boundary of this ellipse and cannot deviate too far—explaining why solutions cluster at axis intercepts.", "This problem illustrates how Diophantine equations connect algebra, geometry, and number theory, offering insight into rational approximations and lattice point problems.", "---", "## Practical Applications", "Finding integer solutions to such equations appears in:", "- Cryptography (lattice-based schemes)\n- Computer graphics (grid-aligned shapes)\n- Optimization (integer programming)\n- Leetcode-style number theory problems", "---", "## Conclusion", "The only integer solutions to ( \frac{x^2}{25} + \frac{y^2}{9} = 1 ) are:", "[\n(x, y) = (5, 0),\ (-5, 0),\ (0, 3),\ (0, -3)\n]", "These represent the four vertex points of the ellipse, where it intersects the integer lattice. Understanding such constrained Diophantine equations strengthens foundational knowledge in number theory and computational mathematics.", "---", "Keywords: integer solutions, Diophantine equation, ( \frac{x^2}{25} + \frac{y^2}{9} = 1 ), lattice points, rational points on ellipses, number theory, Diophantine equations, algebra and geometry."]