Sun–Merca–Tsai Conjecture: Partition function repels perfect powers

Establish that, for every integer n ≥ 2 and every integer k ≥ 2, the partition function p(n) is never equal to a perfect k-th power m^k with m ≥ 2; and determine that, for any fixed integers k ≥ 2 and d ≥ 0, at most finitely many integers n satisfy Δ_k(n) ≤ d, where Δ_k(n) = min_{m ≥ 0} |p(n) − m^k|.

Background

The paper studies perfect-power values of partition functions, focusing on Sun’s conjecture that the unrestricted partition function p(n), for n > 1, never assumes a perfect power value, and on a strengthened form proposed by Merca et al. asserting a quantitative repulsion from perfect powers. Writing Δk(n) = min{m ≥ 0} |p(n) − m^k|, the conjecture has two parts: the absence of exact perfect-power values and finiteness of n for which p(n) lies within a fixed bounded distance d of a k-th power.

The note proves analogous results for truncated partition functions p_B(n), defined by P_B(q) = ∏_{m=1}^{B} (1−q^m)^{-1}, showing perfect-power repulsion when B ≥ 4 and k ≥ 3 with k ∤ (B−1). These results support—but do not settle—the conjecture for p(n) itself, since p_B(n) → p(n) as B → ∞. The proof reduces the problem to finiteness of integral points on superelliptic curves via Siegel’s theorem, leveraging the quasipolynomial structure of p_B(n) on arithmetic progressions.