If p is prime, then let f p denote the Legendre symbol modulo p and let e p be the trivial character modulo p. As usual, let n1

Eighty years ago, Ramanujan conjectured and proved some striking congruences for the partition function modulo powers of 5, 7, and 11. Until recently, only a handful of further such congruences were known. Here we report that such congruences are much more widespread than was previously known, and we describe the theoretical framework that appears to explain every known Ramanujan-type congruence.

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We identify a parameterized family of K 3 surfaces with generic Picard number 19, and we employ elementary methods to determine their local zeta functions. In addition, we explicitly determine those surfaces which are modular.