A note on split dilations defined by higher residues
Jay P. Fillmore · Proceedings of the American Mathematical Society · 1967
where (x/p) is the quadratic residue symbol, has been called a split dilation by W. H. Mills. If ab O and (a/p) = (b/p), it is a permutation of the elements of Fp. Two maps P, Q of Fp into itself are called equivalent if Q(x) =P(x+a)+b for some a, b and all x in Fp. Two split dilations have the property that they are equivalent only when they are equal. In this note we discuss split dilations of a field with q elements where the splitting is defined for any divisor of q1. Such split dilations are equivalent only when they are equal. The attractive conjecture that every permutation is equivalent to some split dilation is shown false by enumerating the split dilations and comparing this with the total number of equivalence classes.