Question: How many lattice points lie on the hyperbola $x^2 - 4y^2 = 100$?

["Optimizing Your Search: How Many Lattice Points Lie on the Hyperbola $x^2 - 4y^2 = 100$?", "When exploring Diophantine equations and geometric structures on the integer lattice, one intriguing question arises: How many lattice points lie on the hyperbola defined by $x^2 - 4y^2 = 100$? Lattice points are points in the coordinate plane where both $x$ and $y$ are integers. Solving this equation for integer solutions uncovers elegant connections between algebra, number theory, and geometry.", "---", "### Understanding the Hyperbola", "The equation $x^2 - 4y^2 = 100$ represents a hyperbola symmetric about both axes and centered at the origin. This form resembles a standard hyperbola but with a scaled $y^2$ term. Rewriting it:", "$$\n\frac{x^2}{100} - \frac{y^2}{25} = 1\n$$", "This confirms it is indeed a hyperbola opening horizontally, with asymptotes $x = \pm 2y$.", "---", "### Goal: Count Integer Solutions $(x, y)$", "We seek all integer pairs $(x, y)$ satisfying:\n$$\nx^2 - 4y^2 = 100\n$$", "This is a Pell-type Diophantine equation. To find lattice points, we exploit factorization:", "$$\nx^2 - 4y^2 = (x - 2y)(x + 2y) = 100\n$$", "Let:\n- $a = x - 2y$\n- $b = x + 2y$", "Then $ab = 100$, and solving for $x$ and $y$ gives:\n$$\nx = \frac{a + b}{2}, \quad y = \frac{b - a}{4}\n$$", "For $x$ and $y$ to be integers, both expressions must yield integers. Thus, we require:\n- $a + b$ even → $a$ and $b$ have the same parity\n- $b - a$ divisible by 4", "---", "### Step 1: Enumerate Factor Pairs of 100", "List all integer factor pairs $(a, b)$ such that $ab = 100$:", "Positive pairs:\n- $(1, 100)$, $(2, 50)$, $(4, 25)$, $(5, 20)$, $(10, 10)$\nNegative pairs:\n- $(-1, -100)$, $(-2, -50)$, $(-4, -25)$, $(-5, -20)$, $(-10, -10)$", "All signs matter, but we aim for same-parity $a, b$ so sum $a + b$ is even.", "Check each pair:", "---", "Positive Pairs:", "1. $(1, 100)$: $a + b = 101$ (odd) → discard\n2. $(2, 50)$: $a + b = 52$ (even), $b - a = 48$, divisible by 4 → valid\n → $x = 52/2 = 26$, $y = 48/4 = 12$ → $(26, 12)$\n3. $(4, 25)$: $a + b = 29$ (odd) → discard\n4. $(5, 20)$: $a + b = 25$ (odd) → discard\n5. $(10, 10)$: $a + b = 20$, $b - a = 0$, divisible by 4 → valid\n → $x = 10$, $y = 0$ → $(10, 0)$", "---", "Negative Pairs:", "1. $(-1, -100)$: sum = -101 (odd) → discard\n2. $(-2, -50)$: sum = -52 (even), $b - a = -48$, divisible by 4 → valid\n → $x = -52/2 = -26$, $y = -48/4 = -12$ → $(-26, -12)$\n3. $(-4, -25)$: sum = -29 (odd) → discard\n4. $(-5, -20)$: sum = -25 (odd) → discard\n5. $(-10, -10)$: sum = -20, $b - a = 0$ → valid\n → $x = -10$, $y = 0$ → $(-10, 0)$", "---", "### Step 2: List All Valid Lattice Points", "From above, valid pairs:\n- $(26, 12)$\n- $(10, 0)$\n- $(-26, -12)$\n- $(-10, 0)$", "Check each satisfies original equation:", "- $26^2 - 4(12)^2 = 676 - 576 = 100$ ✅\n- $10^2 - 4(0)^2 = 100$ ✅\n- $(-26)^2 - 4(-12)^2 = 676 - 576 = 100$ ✅\n- $(-10)^2 - 4(0)^2 = 100$ ✅", "Are there more?", "---", "### Are There More Factorizations?", "We considered all integer divisor pairs of 100. Since 100 has $9$ positive divisors and $9$ negative, total $18$ factor pairs — we’ve checked all with correct parity.", "Note: $(10,10)$ and $(-10,-10)$ give $y=0$, which are valid.", "Also, $(x,y) = (10, 0)$, $(-10, 0)$, $(26,12)$, $(-26,-12)$", "But wait — what about $(x,y) = (-26, 12)$? Let’s test:\n$(x^2 - 4y^2) = 676 - 4(144) = 676 - 576 = 100$ — yes! But $b - a = -10 - (-26) = 16$, $y = 16/4 = 4$, no — earlier we used $b - a = (x+2y) - (x-2y) = 4y$, so $y = 4$, but $x = -26$, $a = x - 2y = -26 - 8 = -34 <br/>\ne -10$", "Wait — correction: When $a = x - 2y$, $b = x + 2y$, so $a = -34$, $b = -16$, $ab = 544 <br/>\ne 100$ — so not in list.", "Our method only includes pairs where $ab = 100$, so all candidate pairs are exhausted.", "But wait — did we miss symmetric sign combinations?", "Try $(a,b) = (-2, -50)$: already done → $x = -26$, $y = -12$", "What about $(a,b) = (-10, -10)$: $x = -10$, $y = 0$ — included", "Is there a pair like $(a,b) = (25, 4)$? Then $ab = 100$, but $a + b = 29$ (odd) → $x = 14.5$, not integer → excluded", "Similarly, $(4,25)$ → sum odd → excluded", "Only pairs where $a, b$ both even (to make $x = (a+b)/2$ integer) and $b - a \equiv 0 \pmod{4}$", "But in negative pairs, $(-26, -10)$: $a = -26$, $b = -10$, $ab = 260 <br/>\ne 100$ — invalid unless $ab=100$", "So no extra pairs.", "---", "### Final List of Lattice Points", "From all valid factorizations with $ab=100$ and correct parity, we have:", "$$\n(x, y) \in { (26, 12), (10, 0), (-26, -12), (-10, 0) }\n$$", "Wait — is $(26, -12)$ a solution?", "Try: $x = 26$, $y = -12$: $x^2 - 4y^2 = 676 - 4(144) = 676 - 576 = 100$ ✅\nNow check $a = x - 2y = 26 - 2(-12) = 26 + 24 = 50$\n$b = x + 2y = 26 + 2(-12) = 26 - 24 = 2$\nThen $ab = 50 \cdot 2 = 100$ — yes! But earlier we didn’t get this pair.", "Why?", "Because in earlier analysis, we assumed $a = x - 2y$, $b = x + 2y$, so $b - a = 4y$, but here $b - a = 2 - 50 = -48$, divisible by 4, and $a + b = 52$, $x = 26$, valid.", "But in our pairing, we required $a \cdot b = 100$, and $a = 50$, $b = 2$: $50 \cdot 2 = 100$ — valid! But why wasn’t it included?", "Because we only considered factor pairs where $a \le b$ in magnitude? No — we listed all combinations.", "Wait: $(a,b) = (50, 2)$: $ab = 100$, $a + b = 52$, $b - a = -48$, divisible by 4 → valid.", "But in earlier step, we only considered:\n- $(2,50)$: $a=2, b=50$\n- $(50,2)$: $a=50, b=2$ — same as ordered pair, but must consider order?", "No — we considered $(a,b)$ as ordered with $ab=100$, so $(50,2)$ is valid and distinct.", "But in our prior list, we used unordered inclusion — but in computation:", "For $(a,b) = (50,2)$:\n$x = (50 + 2)/2 = 26$, $y = (2 - 50)/4 = -48/4 = -12$ → valid\nSo $(26, -12)$ is a separate solution", "Similarly, $( -50, -2 )$: $x = -26$, $y = 12$ → $x = (a+b)/2 = -26$, $y = (-2 - (-50))/4 = 48/4 = 12$ → $(-26,12)$", "Check: $x^2 - 4y^2 = 676 - 576 = 100$ ✅", "So we missed these!", "---", "### Update: Include all orderings", "Factor pairs $(a,b)$ such that $ab=100$:", "List all ordered integer pairs:", "(1,100), (2,50), (4,25), (5,20), (10,10),\n(-1,-100), (-2,-50), (-4,-25), (-5,-20), (-10,-10),\n(100,1), (50,2), (25,4), (20,5), (10,10),\n(-100,-1), (-50,-2), (-25,-4), (-20,-5), (-10,-10) — but duplicates", "Better: all $(a,b)$ with $ab=100$, $a,b \in \mathbb{Z}$", "Total: each divisor pair produces one ordered pair.", "We previously only considered symmetric and positive, but need all 18 ordered pairs? Only 18 divisor combinations? No — infinitely many? No — 100 has finitely many divisors.", "There are 18 ordered integer pairs $(a,b)$ such that $ab=100$:\nFor each positive divisor $d$ of 100, $(d, 100/d)$ and $(-d, -100/d)$, and also $(100/d, d)$, $(-100/d, -d)$ — but only if distinct.", "Actually: number of integer divisors: 9 positive, 9 negative → 18 total values for $a$, each gives unique $b = 100/a$", "So 18 ordered pairs.", "But only those with $a$ and $b$ same parity and $b - a \equiv 0 \pmod{4}$ will give integer $x, y$", "So go back and evaluate all 18?", "But better: since $x = (a+b)/2$, $y = (b - a)/4$, and $a = d$, $b = 100/d$, $d \in \mathbb{Z} \setminus {0}$, we loop over all integer divisors $d$ of 100.", "List all divisors of 100:\n$\pm1, \pm2, \pm4, \pm5, \pm10, \pm20, \pm25, \pm50, \pm100$", "For each $d$, compute:", "- $a = d$\n- $b = 100/d$\n- $x = (a + b)/2$\n- $y = (b - a)/4$\n- Check if $x, y \in \mathbb{Z}$", "---", "### Exhaustive Evaluation:", "1. $d = 1$: $a=1$, $b=100$ → $x = 101/2 = 50.5$ ❌\n2. $d = 2$: $a=2$, $b=50$ → $x=26$, $y = (50-2)/4 = 48/4 = 12$ ✅ → $(26,12)$\n3. $d = 4$: $a=4$, $b=25$ → $x=29/2=14.5$ ❌\n4. $d = 5$: $a=5$, $b=20$ → $x=25/2=12.5$ ❌\n5. $d = 10$: $a=10$, $b=10$ → $x=10$, $y=0$ ✅ → $(10,0)$\n6. $d = 20$: $a=20$, $b=5$ → $x=25/2=12.5$"]