Sparsity of equal higher-dimensional partition counts

Prove that for integers 2<d<e, any equality p_d^n=p_e^m can occur only when d=3.

Background

The authors investigate repetitions among the integers p_dn. Their computational data show many equalities, but in every observed equality with 3≤d<e, the smaller size parameter is d=3. They therefore conjecture that no equality with both size parameters greater than 3 exists, while noting that a general algorithm for all related integral-point problems is unavailable.

References

Fix $2<d<e$. If $p_dn=p_{e}m$ for some $n,m$, then $d=3$.

— Enumeration of partitions via socle reduction  (2501.10267 - Graffeo et al., 17 Jan 2025) in Conjecture 3, Section “Conjectures,” subsection “Sparsity”