A Congruence for the Fermat Quotient Modulo p3.
Khongsit, S.; Saikia, P.K.;
A Congruence for the Fermat Quotient Modulo p3.
Summary: For a prime \(p > 3\) and the Fermat quotient \(q_p(2) = (2^{p-1} -1)/p\), \textit{Z. Sun} [J. Number Theory 128, No. 2, 280--312 (2008; Zbl 1154.11010)] proved that \[ \sum_{k=1}^{p-1} \frac{2^k}{k}+2q_p(2) \equiv -\frac{7}{12}p^2B_{p-3}\pmod{p^3}. \] where \(B_n\) is the \(n\)-th Bernoulli number. In this note, we give an elementary proof of this congruence.
Congruences; primitive roots; residue systems, Bernoulli number, Bernoulli and Euler numbers and polynomials
Congruences; primitive roots; residue systems, Bernoulli number, Bernoulli and Euler numbers and polynomials