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Home » Odd contributions only matter if we have odd primes, but here none → all even - SUpost

Odd contributions only matter if we have odd primes, but here none → all even - SUpost

📅 July 6, 2026 👤 scraface

Jul 06, 2026

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📌 At \( x = \frac{2}{\sqrt{3}} \):

📌 g\left(\frac{2}{\sqrt{3}}\right) = \left(\frac{8}{3\sqrt{3}}\right) - 4\left(\frac{2}{\sqrt{3}}\right) + 2 = \frac{8 - 24 + 6\sqrt{3}}{3\sqrt{3}} = \frac{-16 + 6\sqrt{3}}{3\sqrt{3}}

📌 This is negative, and at \( x = -\frac{2}{\sqrt{3}} \), it's positive. Since \( g(1) = -1 \) and \( g(2) = 2 \), there is a root in \( (1, 2) \), and by symmetry and behavior, another real root exists (confirm via intermediate value theorem or graphing).

📌 Alternatively, use cubic formula or numerical solvers:

📌 Using approximation or a calculator, the real roots are approximately:

📌 One root near \( x \approx 0.542 \)

📌 One root near \( x \approx 1.532 \)

📌 One root near \( x \approx -2.074 \)

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