The fourth power of the Fermat quotient (Q1011457)

For a prime \(p\) and an integer \(a\) with \(p\nmid a\), define the Fermat quotient \[ q_p(a)=\frac{a^{p-1}-1}{p}. \] It is well-known that \[ q_p(2)\equiv-\frac12 G_1(2)\pmod p, \] where \[ G_m(X):=\sum_{i=1}^{p-1}\frac{X^i}{i^m}. \] In [Integers 4, A22, 3 p. (2004; Zbl 1083.11005)], \textit{A. Granville} confirmed a conjecture of Skula that \[ (q_p(2))^2\equiv-G_2(2)\pmod p\quad\text{for}\;p\geq 5. \] Subsequently, \textit{K. Dilcher} and \textit{L. Skula} [Integers 6, Paper A24, 12 p., electronic only (2006; Zbl 1103.11011)] proved that \[ (q_p(2))^3\equiv-3G_3(2)-\frac78B_{p-3}\pmod p\quad\text{for prime}\;p\geq 5. \] In this paper, the authors show that for \(p\geq 7\), \[ (q_p(2))^4\equiv-12G_4(2)+2\sum_{k=0}^{p-4}\binom{p-3}{k}B_kG_{k+3}(2)+\frac74B_{p-3}G_1(2)\pmod p. \]