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Home » 2x + 7 + 5x - 1 + 3x + 4 = 10x + 10 - 500apps

2x + 7 + 5x - 1 + 3x + 4 = 10x + 10 - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 so $ x^6 \equiv 1 \pmod{d(x)} $. Thus, modulo $ d(x) $, all powers of $ x $ reduce modulo 6.

📌 We reduce $ f(x) \equiv (x^2 + 1)^{2025} + 1 \pmod{d(x)} $.

📌 Letâs analyze $ (x^2 + 1)^{2025} \mod d(x) $.

📌 Let $ y = x^2 $, then $ d(x) = y^2 + y + 1 $. Since $ y^3 \equiv 1 \mod (y^2 + y + 1) $, we have $ y^3 \equiv 1 $, so powers of $ y $ reduce modulo 3.

📌 Now $ x^2 \equiv y $, so $ x^2 + 1 \equiv y + 1 $, and

📌 We reduce $ (y + 1)^{2025} $ modulo $ y^2 + y + 1 $. Since $ y^2 \equiv -y - 1 $, we can reduce all powers of $ y $ to linear expressions.

📌 Letâs compute the reduction of $ (y + 1)^{2025} $ modulo $ y^2 + y + 1 $. Letâs denote $ \omega $ as a primitive cube root of unity (so $ \omega^3 = 1 $, $ \omega

📌 e 1 $, and $ \omega^2 + \omega + 1 = 0 $), and note that this polynomial divides $ y^2 + y + 1 $.

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