Some observations and speculations on partitions into $d$-th powers

TL;DR

This paper explores the arithmetic properties and asymptotic behavior of the partition function into d-th powers, presenting new congruences and conjectures based on extensive computational data.

Contribution

It introduces new congruences for partition functions into d-th powers and offers conjectures on their arithmetic behavior based on large-scale computations.

Findings

Proved new congruences involving partitions into d-th powers.

Generated extensive computational data for n up to 10^5 and 10^6.

Formulated conjectures on the arithmetic properties of these partition functions.

Abstract

The aim of this note is to provoke discussion concerning arithmetic properties of function $p_{d}(n)$ counting partitions of an positive integer $n$ into $d$-th powers, where $d\geq 2$. Besides results concerning the asymptotic behavior of $p_{d}(n)$ a little is known. In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers. The second part of the paper has experimental nature and contains questions and conjectures concerning arithmetic behavior of the sequence $(p_{d}(n))_{n\in\N}$. They based on our computations of $p_{d}(n)$ for $n\leq 10^5$ in case of $d=2$, and $n\leq 10^{6}$ for $d=3, 4, 5$.