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Home » \sum_k=1^50 rac1k(k+2). - Leaselab

\sum_k=1^50 rac1k(k+2). - Leaselab

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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Mod 5: \(n^3 \equiv 13 \equiv 3 \pmod{5}\)
Check cubes mod 5:
\(0^3 = 0\)
\(1^3 = 1\)
\(2^3 = 8 \equiv 3\)
\(3^3 = 27 \equiv 2\)
\(4^3 = 64 \equiv 4\)
So only \(n \equiv 2 \pmod{5}\) works.
Now lift to mod 25: Let \(n = 5k + 2\). Plug into \(n^3 \equiv 13 \pmo...
n^3 = (5k+2)^3 = 125k^3 + 3\cdot25k^2\cdot2 + 3\cdot5k\cdot4 + 8 = 125...

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