Sun’s perfect-power conjecture for partition numbers

Prove that for every integer n > 1, the partition number p(n) is not a kth power of an integer for any integer k > 1.

Background

The paper studies whether values of the integer partition function p(n) can be perfect powers. It records Zhi-Wei Sun’s 2013 conjecture that no partition number p(n) with n > 1 is a kth power for any exponent k > 1.

The authors establish the conjecture for a substantial set of n using results of Bennett and Siksek on Diophantine equations of the form x² + qᵃ = yᵏ, but explicitly state that the conjecture itself remains unresolved.

References

Although we are unable to prove this conjecture, we show that recent deep work of Bennett and Siksek [3] on Diophantine equations offers strong evidence supporting its truth.

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section “Sun’s Conjecture”

Conjecture (Sun). If n > 1, then p(n) is not a kth power of an integer for any k > 1. Although we are unable to prove this conjecture, we show that recent deep work of Bennett and Siksek [3] on Diophantine equations offers strong evidence supporting its truth.

— Do perfect powers repel partition numbers?  (2501.03754 - Merca et al., 7 Jan 2025) in Section 1, “Sun’s Conjecture”