Note on Wieferich's congruence for primes \(p\equiv 1\pmod 4\)
For primes \(p \equiv 1 \pmod 4\) it is known that the Euler number \(E_{p-1}\) is divisible by \(p\). In this note the author computes \(E_{p-1}/p\) modulo \(p\). Writing \(p\) in the form \(a^2 + 4b^2\) he in fact shows that \[ E_{p-1}/p + 2Q_2 + 2Q_a + 1/(2a^2) \equiv 0 \pmod p, \] where \(Q_n = (n^{p-1}-1)/p\), the Fermat quotient. The rather involved computation is based on the author's previous work, particularly on a result about the Gauss sum attached to a character mod \(p\) of order 4 [Acta Arith. 85, No. 4, 377-388 (1998; Zbl 0912.11041)]. Of special interest is the case that \(Q_2 \equiv 0 \pmod p\), in other words, \(p\) is a Wieferich prime.
scientific article; zbMATH DE number 1416680