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Published in last 50 years
We shall prove a sequence of congruences modulo odd primes p which can be viewed as generalizations of a congruence first proved by Zhi-Wei Sun in 1995.
In this note, we give a simple and elementary proof of the following curious congruence which was established by Zhi-Wei Sun: \[ ∑ k = 1 ( p − 1 ) / 2 1 k ⋅ 2 k ≡ ∑ k = 1 [ 3 p / 4 ] ( − 1 ) k − 1 k ( m o d p ) . \sum ^{(p-1)/2}_{k=1}\frac {1}{k\cdot 2^k}\equiv \sum ^{[3p/4]}_{k=1} \frac {(-1)^{k-1}}{k}\quad (\mathrm {mod}\,p). \]