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Some observations and speculations on partitions into $d$-th powers
Number Theory · Submitted 10 Feb 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers.
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
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The abstract, sentence by sentence
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 $d=3, 4, 5$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $d\geq 2$. | contribution 0.55 | 0.39 | 0.06 | asserted 0.80 |
| s1 | Besides results concerning the asymptotic behavior of $p_{d}(n)$ a little is known. | gap 0.85 | 0.06 | 0.04 | asserted 0.99 |
| s2 | In the first part of the paper, we prove certain congruences involving functions counting various types of partitions into $d$-th powers. | contribution 0.93 | 0.95 | 0.14 | asserted 1.00 |
| s3 | The second part of the paper has experimental nature and contains questions and conjectures concerning arithmetic behavior of the sequence $(p_{d}(n))_{n\in\N}$. | contribution 0.98 | 0.85 | 0.06 | asserted 0.85 |
| s4 | 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 $d=3, 4, 5$. | method 0.98 | 0.95 | 0.04 | asserted 0.99 |
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The paper
Original 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 $d=3, 4, 5$.
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