Polynomial-time computation of the squarefree indicator μ^2(n)

Determine whether there exists a polynomial-time algorithm that, on input a positive integer n, computes the squarefree indicator function μ^2(n).

Background

The paper investigates whether small transformer models can predict values of the Möbius function μ(n) and the squarefree indicator μ^2(n) from Chinese Remainder Theorem encodings of n. In setting expectations about computational hardness, the authors note that the most direct approach to computing μ(n) or μ^2(n) is to factor n, for which no substantially faster general algorithm is known.

They reference prior work highlighting the computational complexity landscape: Adleman and McCurley observe that it is unknown whether μ^2(n) can be computed in polynomial time, Shallit and Shamir show μ(n) is polynomial-time computable given an oracle for the divisor-count function d(m) for a specific m, and Booker–Hiary–Keating give a GRH-conditional subexponential algorithm for μ^2(n) whose running time is conjecturally slower than current factoring algorithms but independent of factorization. This situates the question of a polynomial-time algorithm for μ^2(n) as a fundamental open problem in computational number theory.