PORC function

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Definition

A function f on an infinite subset S of the natural numbers is termed a Polynomial On Residue Classes function or PORC function if there exists a natural number m and polynomials f_0, f_1, \dots, f_{m-1} such that if n \equiv a \pmod m with n \in S and 0 \le a \le m - 1, then f(n) = f_a(n).

In other words, the function f behaves like a polynomial on each of the residue classes modulo m.

PORC functions are also called quasipolynomials, though that term has many other meanings in other contexts.

Facts

  • Higman's PORC conjecture: For a fixed natural number n, define f(p,n) for a prime p as the number of isomorphism groups of order p^n. Higman conjectured that for any fixed n, f(p,n) is a PORC function of p.