Minimal sufficient sets of colors and minimum number of colors

In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6 for p \geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.