So we solve \( 9x^2 + 16y^2 = 144 \) in integers with \( |x| \leq 4 \), \( |y| \leq 3 \).
["Solving the Diophantine Equation ( 9x^2 + 16y^2 = 144 ) for Integer Solutions\nWith Constraints ( |x| \leq 4 ) and ( |y| \leq 3 )", "---", "Understanding the Equation", "The equation ( 9x^2 + 16y^2 = 144 ) represents a symmetric quadratic Diophantine equation. Our goal is to find all integer pairs ((x, y)) satisfying this equation while respecting the bounds ( |x| \leq 4 ) and ( |y| \leq 3 ). Restricting ( x ) and ( y ) to these ranges simplifies the search for solutions significantly.", "---", "### Step 1: Rewrite the Equation", "Start by dividing the entire equation by 144 to normalize:", "[\n\frac{9x^2}{144} + \frac{16y^2}{144} = 1 \quad \Rightarrow \quad \frac{x^2}{16} + \frac{y^2}{9} = 1\n]", "This resembles the standard form of an ellipse:\n[\n\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \ ext{with } a = 4, b = 3\n]", "But since we seek only integer solutions within bounded domains, we solve directly by substitution.", "---", "### Step 2: Use Given Constraints", "We are to find integer pairs ((x, y)) such that:", "- ( |x| \leq 4 \Rightarrow x \in {-4, -3, -2, -1, 0, 1, 2, 3, 4} )\n- ( |y| \leq 3 \Rightarrow y \in {-3, -2, -1, 0, 1, 2, 3} )", "For each integer ( x ) in ([-4, 4]), compute ( 9x^2 ), then solve for ( y^2 ):", "[\n16y^2 = 144 - 9x^2 \quad \Rightarrow \quad y^2 = \frac{144 - 9x^2}{16}\n]", "We require ( y^2 ) to be a non-negative perfect square.", "---", "### Step 3: Compute Possible ( y^2 ) for Each ( x )", "We evaluate each ( x ) and check whether ( y^2 = \frac{144 - 9x^2}{16} ) is:", "- An integer\n- A perfect square\n- Within ( 0 \leq y^2 \leq 9 ) (since ( |y| \leq 3 \Rightarrow y^2 \leq 9 ))", "Let’s compute:", "| ( x ) | ( x^2 ) | ( 9x^2 ) | ( 144 - 9x^2 ) | ( y^2 = \frac{144 - 9x^2}{16} ) | Integer? | Perfect Square? | Valid ( y ) range? |\n|--------|----------|------------|------------------|----------------------------------|----------|-----------------|------------------------|\n| -4 | 16 | 144 | 0 | 0 | Yes | ( 0^2 ) | ( y = 0 ) (valid)\n| -3 | 9 | 81 | 63 | ( 63/16 = 3.9375 ) | No | — | —\n| -2 | 4 | 36 | 108 | ( 108/16 = 6.75 ) | No | — | —\n| -1 | 1 | 9 | 135 | ( 135/16 = 8.4375 ) | No | — | —\n| 0 | 0 | 0 | 144 | ( 144/16 = 9 ) | Yes | ( 3^2 ) | ( y = \pm3 ) (valid)\n| 1 | 1 | 9 | 135 | ( 135/16 = 8.4375 ) | No | — | —\n| 2 | 4 | 36 | 108 | ( 108/16 = 6.75 ) | No | — | —\n| 3 | 9 | 81 | 63 | ( 63/16 = 3.9375 ) | No | — | —\n| 4 | 16 | 144 | 0 | ( y^2 = 0 ) → ( y = 0 ) | Yes | ( 0^2 ) | Valid", "Now check ( x = 0 ) and ( x = \pm4 ) — both yield clean solutions.", "---", "### Step 4: List All Valid Integer Solutions", "From the table, the only values yielding integer and perfect square ( y^2 ) within bounds are:", "- ( (x, y) = (-4, 0) ): ( y = 0 ), ( y^2 = 0 )\n- ( (x, y) = (0, \pm3) ): ( x = 0 ), ( y^2 = 9 \Rightarrow y = \pm3 )\n- ( (x, y) = (\pm4, 0) ): ( y = 0 )", "Now verify each:", "- For ( (x, y) = (-4, 0) ):\n ( 9(16) + 16(0) = 144 ) ✅", "- For ( (0, 3) ), ( (0, -3) ):\n ( 9(0) + 16(9) = 144 ) ✅\n ( 9(16) + 0 = 144 ) ✅", "No other ( x ) yields valid ( y ).", "---", "### Final Answer: Integer Solutions with Constraints", "The integer solutions to ( 9x^2 + 16y^2 = 144 ) with ( |x| \leq 4 ) and ( |y| \leq 3 ) are:", "[\n\boxed{\n\begin{aligned}\n&(-4, 0), \\n&(0, 3),\quad (0, -3), \\n&(4, 0)\n\end{aligned}\n}\n]", "These represent four distinct integer solutions satisfying both the equation and the bounded domain.", "---", "### Why This Problem Matters", "Solving Diophantine equations within bounded regions helps in algorithmic filtering, cryptography, and Diophantine approximation studies. This problem is a small but instructive example when teaching integer solutions under constraints — essential for coding theory and computational number theory.", "---", "Keywords:\nDiophantine equation, integer solutions, ( 9x^2 + 16y^2 = 144 ), ( |x| \leq 4 ), ( |y| \leq 3 ), bounded integer solutions, algebraine constraints, number theory problem.", "Meta Description:\nFind all integer pairs ( (x, y) ) satisfying ( 9x^2 + 16y^2 = 144 ) with ( |x| \leq 4 ) and ( |y| \leq 3 ). Complete solution with verification and constraints.", "---", "Explore more: Try replacing 144 with other values or relax constraints to see how solution counts increase."]
