This paper presents a novel approach at the intersection of machine learning
and number theory, focusing on the classification of prime and non-prime
numbers. At the core of our research is the development of a highly sparse
encoding method, integrated with conventional neural network architectures.
This combination has shown promising results, achieving a recall of over 99\%
in identifying prime numbers and 79\% for non-prime numbers from an inherently
imbalanced sequential series of integers, while exhibiting rapid model
convergence before the completion of a single training epoch. We performed
training using $10^6$ integers starting from a specified integer and tested on
a different range of $2 \times 10^6$ integers extending from $10^6$ to $3
\times 10^6$, offset by the same starting integer. While constrained by the
memory capacity of our resources, which limited our analysis to a span of
$3\times10^6$, we believe that our study contribute to the application of
machine learning in prime number analysis. This work aims to demonstrate the
potential of such applications and hopes to inspire further exploration and
possibilities in diverse fields.

该研究论文介绍了一种机器学习和数论相交的新方法，关注的是素数和非素数的分类。研究的核心是开发了一种高度稀疏的编码方法，与传统的神经网络结构相结合。这种组合已经显示出有希望的结果，在识别自然存在不平衡的整数序列中的素数方面，召回率达到 99％以上，对于非素数为 79％，而且在完整的单次训练迭代前就展现出快速的模型收敛。我们进行了 $10^6$ 个整数的训练，从指定的整数开始，并在从 $10^6$ 到 $3 	imes 10^6$ 的不同范围的 $2 	imes 10^6$ 个整数上进行测试，其起始整数相同。虽然受到我们资源的内存容量的限制，将分析限制在 $3 	imes 10^6$ 的范围内，但我们认为这项研究对机器学习在素数分析中的应用做出了贡献。这项工作旨在展示这种应用的潜力，并希望在不同领域中激发进一步的探索和可能性。

探索质数分类：以稀疏编码实现高召回率和快速收敛

Exploring Prime Number Classification: Achieving High Recall Rate and  Rapid Convergence with Sparse Encoding

Log-loss (also known as cross-entropy loss) metric is ubiquitously used
across machine learning applications to assess the performance of
classification algorithms. In this paper, we investigate the problem of
inferring the labels of a dataset from single (or multiple) log-loss score(s),
without any other access to the dataset. Surprisingly, we show that for any
finite number of label classes, it is possible to accurately infer the labels
of the dataset from the reported log-loss score of a single carefully
constructed prediction vector if we allow arbitrary precision arithmetic.
Additionally, we present label inference algorithms (attacks) that succeed even
under addition of noise to the log-loss scores and under limited precision
arithmetic. All our algorithms rely on ideas from number theory and
combinatorics and require no model training. We run experimental simulations on
some real datasets to demonstrate the ease of running these attacks in
practice.

本文探讨了如何从单个（或多个）log-loss 得分中推断数据集的标签，我们通过引入数论和组合数学的思想提出了对抗性侦查方法进行实验模拟，可成功推断数据集的标签。

从对数损失分数中推断标签

Label Inference Attacks from Log-loss Scores

In this work, we introduce $\beta$-expansion, a notion borrowed from number
theory, as a theoretical framework to study fast construction of polar codes
based on a recursive structure of universal partial order (UPO) and
polarization weight (PW) algorithm. We show that polar codes can be recursively
constructed from UPO by continuously solving several polynomial equations at
each recursive step. From these polynomial equations, we can extract an
interval for $\beta$, such that ranking the synthetic channels through a
closed-form $\beta$-expansion preserves the property of nested frozen sets,
which is a desired feature for low-complex construction. In an example of AWGN
channels, we show that this interval for $\beta$ converges to a constant close
to $1.1892 \approx 2^{1/4}$ when the code block-length trends to infinity. Both
asymptotic analysis and simulation results validate our theoretical claims.

本文提出 $eta$-expansion 作为理论框架来研究基于 UPO 和 PW 算法的极化码的快速构建，并在 AWGN 信道的示例中展示了这个框架的可行性和有效性

$β$- 展开式：极化码的快速递归构造的理论框架

$β$-expansion: A Theoretical Framework for Fast and Recursive  Construction of Polar Codes

We survey results on the pebbling numbers of graphs as well as their
historical connection with a number-theoretic question of Erd\H os and Lemke.
We also present new results on two probabilistic pebbling considerations, first
the random graph threshold for the property that the pebbling number of a graph
equals its number of vertices, and second the pebbling threshold function for
various natural graph sequences. Finally, we relate the question of the
existence of pebbling thresholds to a strengthening of the normal property of
posets, and show that the multiset lattice is not supernormal.

本文主要研究了关于图的表示数及其与数论问题的历史联系。此外，我们还研究了概率方面的新结果，以及它们与自然图序列的表示函数的联系。最后，我们将表示阈值的存在问题与偏序的正常性加强联系起来，并证明了多重集格子不是超正常的。