PORC function

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 $0\leq a\leq m-1$, 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 $p$.