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Home » Is there (0,0,4)? No. - Abbey Badges

Is there (0,0,4)? No. - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 So \( rac{a + b}{a - b} = rac{3 \pm \sqrt{5}}{2} \). Letâs take one root (both valid due to symmetry). Set:

📌 rac{a + b}{a - b} = rac{3 + \sqrt{5}}{2}

📌 2(a + b) = (3 + \sqrt{5})(a - b)

📌 2a + 2b = (3 + \sqrt{5})a - (3 + \sqrt{5})b

📌 2a - (3 + \sqrt{5})a + 2b + (3 + \sqrt{5})b = 0

📌 \Rightarrow a(-1 - \sqrt{5}) + b(5 + \sqrt{5}) = 0

📌 Solve for \( rac{a}{b} \):

📌 a b ( -\overline{1 + \sqrt{5}} + (5 + \sqrt{5}) ) = 0 \Rightarrow -rac{a}{b} - \sqrt{5} \cdot rac{a}{b} + 5 + \sqrt{5} = 0

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