Eisenstein likened the first two of these congruences to properties of logarithms. Euclidean geometry corresponds to the subgroup of congruences. Alhazen solved problems involving congruences using what is now called Wilson's theorem. Unlike linear equations over the reals, linear congruences may have zero, one or several solutions. Wieferich primes can be defined by other equivalent congruences. He proved many congruences for these numbers, such as for primes . The latter condition rules out Euclidean congruences such as simple translation and rotation. Generating functions also have other uses in proving congruences for their coefficients. There is also an optical (or null) version of Raychaudhuri's equation for null geodesic congruences. Addition and multiplication are compatible with congruences. Let formula 35 be the set of congruences on the algebra formula 1. An algebra with no other congruences is called "simple". A generalization of these congruences goes by the name of -adic continuity. In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function "p"("n"). Suppose that and are both solutions to all the congruences.