Affine transformations of finite vector spaces with large orders or few cycles

Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least pd/4, and apply this classification to determine the finite primitive permutation groups of affine type, and of degree n, that contain a permutation of order at least n/4. Using this result we obtain a classification of finite primitive permutation groups of affine type containing a permutation with at most four cycles.