A congruence for primes

Abstract

With the help of the Pell sequence we obtain the following new congruence for odd primes: \[ ∑ k = 1 ( p − 1 ) / 2 1 k ⋅ 2 k ≡ ∑ k = 1 [ 3 p / 4 ] ( − 1 ) k − 1 k ( mod p ) . \sum \limits _{k = 1}^{(p - 1)/2} {\frac {1}{{k \cdot {2^k}}} \equiv \sum \limits _{k = 1}^{[3p/4]} {\;\frac {{{{( - 1)}^{k - 1}}}}{k}} \quad \pmod p.} \]