A Generalization of the Goresky–Klapper Conjecture, Part II
Todd Cochrane, Michael J. Mossinghoff, Christopher G. Pinner, C. J. Richardson · Experimental Mathematics · 2019
Suppose that f(x)=Axk mod p is a permutation of the least residues mod p. With the exception of the maps f(x) = Ax and Ax(p+1)/2 mod p we show that for fixed n≥2 the image of each residue class mod n contains elements from every residue class mod n, once p is sufficiently large. If f(x) = Ax mod p, then for each p and n there will be exactly (1+o(1))6π2n2 readily describable values of A for which the image of some residue class mod n misses at least one residue class mod n, even when p is large relative to n. A similar situation holds for f(x)=Ax(p+1)/2 mod p.