A CONGRUENCE FOR FERMAT QUOTIENT

Abstract

Let \(p\) be a prime, and let \(q_p(2)=(2^{p-1}-1)/p\) be the Fermat quotient of \(p\) to base 2. In this paper we prove that for any prime \(p>3\)
\[
q_p(2)\equiv\frac{(-1)^{\lfloor p/3\rfloor-\lfloor p/6\rfloor}3^{(p-1)/2}-1}{p}
-\frac{(-1)^{\lfloor p/3\rfloor-\lfloor p/6\rfloor}3^{(p-3)/2}}{2}
\sum_{k=(p+1)/2}^{p-1}\frac{(-3)^k}{k}\pmod p,
\]
where \(\lfloor a\rfloor\) denotes the greatest integer not exceeding \(a\).