A polynomial \(g(x)\) with integral coefficients is called a permutation polynomial \(\pmod n\), \(n > 1\), provided the mapping \(i \to g(i) \pmod n\), \(i = 1, 2, \dots, n\), is a permutation of the residue classes \(\pmod n\). Let \(M(g(x))\) denote the set of all \(n > 1\) for which \(g(x)\) is a permutation polynomial \(\pmod n\). Let \(P(g(x))\) denote the set of primes contained in \(M(g(x))\) and let \(S(g(x))\) denote the set of primes all of whose powers are contained in \(M(g(x))\). We recall that \(g(x)\) is a permutation polynomial \(\pmod{p^e}\), \(e=1, 2, 3, \dots\), provided \(g(x)\) is a permutation polynomial \(\pmod n\) and \(g'(a)\not\equiv 0\pmod p\) for all integral \(a\). The following results are proved:
1. Let \(R = \{p_1, \dots, p_r)\), \(S = \{q_1, \dots, q_s)\) denote two finite sets of primes, \(R\cap S = \emptyset\). Then there exist normalized polynomials \(g(x)\) of infinitely many distinct degrees such that \(P(g(x)) = R\cup S\), \(S(g(x)) = S\).
2. Let \(n > 1\) and let \(n\) be a permutation of \(1, 2,\dots, n\) that can be represented by a polynomial \(\pmod n\). Let \(n = p_1\dots p_rq_1^{e_1}\dots q_s^{e_s}\), (all \(e_j> 1)\), be the canonical factorization of \(n\). Then \(n\) can be represented by a normalized polynomial \(g(x)\) such that
\[ P(g(x)) = \{p_1, \dots, p_r, q_1, \dots, q_s\},\quad S(g(x)) = \{q_1, \dots, q_s\}. \] Indeed the degree of \(g(x)\) can be assumed to be an arbitrary prime \(\geq n+3\).