DICKSON POLYNOMIALS OF THE 2ND KIND THAT ARE PERMUTATIONS

It is known that the Dickson polynomial of the second kind SIGMA(i=0)[n/2] (n-i\i)(-1)(i)x(n-2i) permutes the elements of the finite prime field F(p) (p odd) when n + 1 = +/- 2 to each of the moduli p, 1/2(p - 1) and 1/2(p + 1). Based on numerical evidence it has been conjectured that these congruences are necessary for the polynomial to permute F(p). The conjecture is established here by a new method.