201103.3907v1 On Lerch´s formula for the Fermat quotient

This paper explores some previously-unrecognized consequences of Lerch´s 1905 formula for the Fermat quotient, with special attention to the sums which he introduced in this context. A generalization of his result is proved, and a new proof given of a sharpened result by Skula (2008). We also sharpen the criteria given by Emma Lehmer in 1938 for a Wieferich prime t..