A Generalization of the Goresky–Klapper Conjecture, Part II. Issue 2 (21st April 2021) - Cochrane, Todd; Mossinghoff, Michael J.; Pinner, Chris; Richardson, C. J.
Abstract: Suppose that f ( x ) = A x k mod p is a permutation of the least residues mod p . With the exception of the maps f ( x ) =  Ax and A x ( 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 ) ) 6 π 2 n 2 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 ) = A x ( p + 1 ) / 2 mod p .