  We study generalised core partitions arising from affine Grassmannian
elements in arbitrary Dynkin type. The corresponding notion of size is given by
the atomic length in the sense of [CLG22]. In this paper, we first develop the
theory for extended affine Weyl groups. In a series of applications, we give
some remarkable parametrisations of the solutions of certain Diophantine
equations resembling Pell's equation, by refining the results of [BN22] and
[Alp14], and generalising them to further types.
