We analyze modulo small values. First, note that $x^2 \geq 144$, so $|x| \geq 12$. Try small integer values of $y$ and check if $25x^2 = 144y^2 + 3600$ yields a perfect square for $x^2$.

["Analyzing Modulo Small Values: Solving $25x^2 = 144y^2 + 3600$ with $|x| \geq 12$", "When tackling Diophantine equations or modular arithmetic challenges, analyzing the problem through constraints and systematic testing can reveal hidden solutions. One powerful technique is reducing the equation modulo small values, leveraging properties like congruences and divisibility to simplify search space.", "This analysis focuses on the equation:\n[\n25x^2 = 144y^2 + 3600\n]\nunder the condition that ( |x| \geq 12 ). We begin by deducing a necessary constraint:\n[\nx^2 \geq 144 \quad \Rightarrow \quad |x| \geq 12\n]\nThis boundary sets a practical lower limit for integer testing.", "Next, reduce the equation modulo small integers to narrow possible values of ( y ) and eliminate nonviable candidates quickly. A particularly insightful modulus here is ( n = 5 ), since the coefficient 25 appears prominently.", "### Step 1: Reduce modulo 5\nReduce both sides modulo 5:\n[\n25x^2 \equiv 0 \pmod{5} \quad \ ext{(because 25 is divisible by 5)}\n]\n[\n144y^2 + 3600 \equiv 144y^2 \pmod{5}\n]\nNote that ( 144 \equiv 4 \pmod{5} ) and ( 3600 \equiv 0 \pmod{5} ), so:\n[\n144y^2 \equiv 4y^2 \pmod{5}\n]\nThus, the congruence becomes:\n[\n4y^2 \equiv 0 \pmod{5} \quad \Rightarrow \quad y^2 \equiv 0 \pmod{5}\n]\nThis implies that ( 5 \mid y^2 ), so ( 5 \mid y ). Therefore, ( y ) must be divisible by 5.", "Rewriting ( y = 5k ) for integer ( k ), substitute into the original equation:\n[\n25x^2 = 144(25k^2) + 3600 \quad \Rightarrow \quad 25x^2 = 3600k^2 + 3600\n]\nDivide both sides by 25:\n[\nx^2 = 144k^2 + 144 = 144(k^2 + 1)\n]\nHence:\n[\nx^2 = 144(k^2 + 1) \quad \Rightarrow \quad x = 12\sqrt{k^2 + 1}\n]\nFor ( x ) to be an integer, ( \sqrt{k^2 + 1} ) must be rational — and since it's an integer ratio, ( k^2 + 1 ) must be a perfect square.", "Let ( k^2 + 1 = m^2 ) for some positive integer ( m ). Then:\n[\nm^2 - k^2 = 1 \quad \Rightarrow \quad (m - k)(m + k) = 1\n]\nThe only integer solution to this is ( m = 1, k = 0 ), since factors of 1 are ( (1,1) ) or ( (-1,-1) ).", "Thus, the only valid integer solution from this congruence analysis is ( k = 0 \Rightarrow y = 0 ), giving:\n[\nx^2 = 144(0 + 1) = 144 \Rightarrow x = \pm 12\n]\nBut check whether ( y = 0 ) satisfies the original equation:\n[\n25x^2 = 144(0)^2 + 3600 \Rightarrow 25x^2 = 3600 \Rightarrow x^2 = 144\n]\nYes — yields ( x = \pm 12 ), satisfying ( |x| = 12 \geq 12 ).", "### Step 2: Verify if smaller ( k <br/>\neq 0 ) could yield valid ( x^2 )\nWe assumed ( k \in \mathbb{Z} \setminus {0} ). But from the Diophantine condition ( m^2 - k^2 = 1 ), no other integer solutions exist, so no other ( k ) gives integer ( x ).", "### Step 3: Confirm small values near boundary — testing modulo patterns\nEven though our modulo 5 reduction gave a tight constraint, consider checking small ( y ) near multiples of 5 (since ( y \equiv 0 \pmod{5} )) to ensure no near-misses or overlooked solutions.", "Try ( y = 5 ):\n[\n25x^2 = 144(25) + 3600 = 3600 + 3600 = 7200 \Rightarrow x^2 = 7200 / 25 = 288 \quad (\ ext{not a square})\n]\n( x^2 = 288 \Rightarrow x \approx 16.97 ), not integer.", "Try ( y = 10 ):\n[\n25x^2 = 144(100) + 3600 = 14400 + 3600 = 18000 \Rightarrow x^2 = 720 \quad (\ ext{not square})\n]\nTry ( y = 15 ):\n[\n25x^2 = 144(225) + 3600 = 32400 + 3600 = 36000 \Rightarrow x^2 = 1440 \quad (\ ext{not square})\n]\nTry ( y = 20 ):\n[\n25x^2 = 144(400) + 3600 = 57600 + 3600 = 61200 \Rightarrow x^2 = 2448 \quad (\ ext{not square})\n]\nAll fail to yield perfect squares, supporting our earlier deduction that no solutions exist for small ( |y| ).", "### Final Thoughts\nAnalyzing small values modulo small integers is a powerful strategy in solving constrained Diophantine equations. Here, reducing modulo 5 uncovered that ( y ) must be divisible by 5. Further analysis showed only ( y = 0 ) yields an integer ( x ) with ( |x| \geq 12 ). No other values under ( k <br/>\neq 0 ) produce valid squares, confirming the uniqueness of this solution under the given bound.", "This method demonstrates how modular analysis can drastically reduce the search space, transforming an algebraic problem into a clean number-theoretic constraint.", "---", "Key Takeaways:\n- Use congruences (e.g., mod 5) to impose divisibility conditions early.\n- Substitution via modular constraints can uncover necessary forms for variables.\n- Verification by testing near solutions ensures no edge cases are missed.\n- Small value analysis combined with modular insights accelerates insight in number theory problems."]