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This paper explores some previously-unrecognized consequences of Lerch's 1905formula for the Fermat quotient, with special attention to the sums which heintroduced in this context. A generalization of his result is proved, and a newproof given of a sharpened result by Skula (2008). We also sharpen the criteriagiven by Emma Lehmer in 1938 for a Wieferich prime to be simultaneously aMirimanoff prime.

On Lerch's formula for the Fermat quotient