Let $K$ be an alternating knot of prime determinant $p$. Consider a minimal diagram, $D$, of $K$ i.e., a diagram of $K$ with least number of arcs. The Kauffman-Harary conjecture states that a $p$-coloring of $D$ assigns different colors to different arcs of any of its minimal diagrams. This conjecture has been proved to be true for rational knots and for Montesinos knots. In this talk we will report on recent work on calculating numbers of colorings and minima of colors without the restriction to minimal diagrams. This is joint work with Louis H. Kauffman.