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Home » But this expression is **unbounded** on $ (0, \frac{\pi - Crosslake

But this expression is unbounded on $ (0, \frac{\pi - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Each such pair yields an integer solution. Therefore, there are 16 lattice points.

📌 But note: each solution \((x, y)\) is counted once for each factor pair. Since \( (a,b) \) and \( (b,a) \) yield potentially different \((x,y)\), but in our setup \(a = x - y\), \(b = x + y\), so order matters in assignment. However, fixing \(a, b\) as both even and same parity already covers all valid factorizations.

📌 Wait: we set \(ab = 2024\), and we require both \(a\) and \(b\) even.

📌 Total factor pairs (positive and negative) where both are even: Since 2024 is divisible by 4, and all factorizations into two evens are valid.

📌 But from the substitution, \(m n = 506\), and 506 is even, but not divisible by higher powers — all its factor pairs consist of one even and one odd? No:

📌 = 2 × 253, and 253 is odd. So in the factor pairs of 506, one factor (the first in a pair) is even, the other odd — unless both are even. But 506 is not divisible by 4?

📌 Wait: 506 ÷ 2 = 253, so 506 ≡ 2 mod 4. So it has exactly one factor of 2. Therefore, in any factor pair \((m, n)\), one is even, one is odd — so \(m\) and \(n\) have alternating parity. But we need both \(m\) and \(n\) even for \(a = 2m\), \(b = 2n\) both even, but \(mn = 506\), which has only one factor of 2, so in any factorization, one is odd, one is even.

📌 Therefore, \(a = 2m\) is even, \(b = 2n\) is even, but \(m\) and \(n\) have opposite parity — so \(a + b = 2(m + n)\), \(b - a = 2(n - m)\) — wait, no:

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