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Large number factorization is not only the most critical entry point for Rivest–Shamir–Adleman (RSA) security analysis, but also the most direct means of attacking the asymmetric encryption algorithm RSA. In this paper, the factorization methods of large numbers are summarized and analysed: classical integer factoring algorithms, Shor’s circuit model algorithm, quantum adiabatic methods (integer factorization based on a quantum nuclear magnetic resonance (NMR) platform and D-Wave quantum annealing), and hybrid quantum-classical computing. Finally, the feasibility of integer factorization based on quantum adiabatics is discussed. In this paper, quantum annealing is regarded as a quantum attack method that is completely different from the famous Shor algorithm, and the potential of D-Wave factorization of large numbers to crack RSA cryptography is verified, which provides a new idea for a quantum attack on RSA public key cryptography.

Before Palm Pilots and iPods, PCs and laptops, the term \"computer\" referred to the people who did scientific calculations by hand. These workers were neither calculating geniuses nor idiot savants but knowledgeable people who, in other circumstances, might have become scientists in their own right. When Computers Were Human represents the first in-depth account of this little-known, 200-year epoch in the history of science and technology. Beginning with the story of his own grandmother, who was trained as a human computer, David Alan Grier provides a poignant introduction to the wider world of women and men who did the hard computational labor of science. His grandmother's casual remark, \"I wish I'd used my calculus,\" hinted at a career deferred and an education forgotten, a secret life unappreciated; like many highly educated women of her generation, she studied to become a human computer because nothing else would offer her a place in the scientific world. The book begins with the return of Halley's comet in 1758 and the effort of three French astronomers to compute its orbit. It ends four cycles later, with a UNIVAC electronic computer projecting the 1986 orbit. In between, Grier tells us about the surveyors of the French Revolution, describes the calculating machines of Charles Babbage, and guides the reader through the Great Depression to marvel at the giant computing room of the Works Progress Administration. When Computers Were Human is the sad but lyrical story of workers who gladly did the hard labor of research calculation in the hope that they might be part of the scientific community. In the end, they were rewarded by a new electronic machine that took the place and the name of those who were, once, the computers.

We show that for all$m,k,r\\in \\mathbb{N}$, there is an$n\\in \\mathbb{N}$such that whenever$L$is a Latin square of order$m$and the Cartesian product$L^{n}$of$n$copies of$L$is$r$-coloured, there is a monochrome Latin subsquare of$L^{n}$, isotopic to$L^{k}$. In particular, for every prime$p$and for all$k,r\\in \\mathbb{N}$, there is an$n\\in \\mathbb{N}$such that whenever the multiplication table$L({\\mathbb{Z}_{p}}^{n})$of the group${\\mathbb{Z}_{p}}^{n}$is$r$-coloured, there is a monochrome Latin subsquare of order$p^{k}$. On the other hand, we show that for every group$G$of order$\\leq 15$, there is a 2-colouring of$L(G)$without a nontrivial monochrome Latin subsquare.

In this paper, we prove the isomorphic criterion theorem for (n+2)-dimensional n-Lie algebras, and give a complete classification of (n + 2)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.

Memorizing the multiplication table is a major challenge for elementary school students: there are many facts to memorize, and they are often similar to each other, which creates interference in memory. Here, we examined whether learning would improve if the degree of interference is reduced, and which memory processes are responsible for this improvement. In a series of 16 short training sessions over 4 weeks, first-grade children learned 16 multiplication facts—4 facts per week. In 2 weeks the facts were dissimilar from each other (low interference), and in 2 control weeks the facts were similar (high interference). Learning in the low-similarity, low-interference weeks was better than in the high-similarity weeks. Critically, this similarity effect originated in the specific learning context, i.e., the grouping of facts to weeks, and could not be explained as an intrinsic advantage of certain facts over others. Moreover, the interference arose from the similarity between facts in a given week, not from the similarity to previously learned facts. Similarity affected long-term memory—its effect persisted 7 weeks after training has ended; and it operated on long-term memory directly, not via the mediation of working memory. Pedagogically, the effectiveness of the low-interference training method, which is dramatically different from currently used pedagogical methods, may pave the way to enhancing how we teach the multiplication table in school.

Among the bamboo slips dating to the Warring States period(475-221 BCE),stored at Tsinghua University,there is an aggregate of 21 specially shaped slips that are painted with red lines on the obverse side.Numbers are written on 20 slips in accordance with certain rules,and on one slip without numbers there are 20 round holes and residues of ribbon.These slips,after being compiled as a whole,serve as a practical calculator in the form of a table,which is designated as Suan Biao算表(the multiplication table).In this paper,the structure and functions of the Suan Biao are discussed by comparing it with other ancient multiplication tables.Specifically,it is noted that the core of the Suan Biao is a multiplication table comprising the numbers 9 to 1 and their products 81 to 1;the other parts are an extension of the core.The Suan Biao employs the decimal place value system and manifested the author’s knowledge of the commutative law of multiplication,the distributive law of multiplication over addition,and fractions.Not only is this table directly applicable to the multiplication of two-digit numbers,but also to division.In addition,certain computations regarding 1/2 or fractions containing 1/2 can be conducted using this table,and it may also be used for extracting roots.The Suan Biao is of great significance for research on ancient mathematics,and its discovery is enormously valuable for studies on arithmetic methods and computational tools in early China.

This study analyses the linguistic features of Korean language in learning multiplication. Ancient Korean multiplication tables, Gugudan, as well as mathematics textbooks and teacher’s guides from South Korea were examined for the instruction of multiplication in second grade. Our findings highlight the uniqueness of the grammatical features of numbers, the syntax of multiplication tables, the simplicity of language of multiplication in Korean language, and also the complexities and ambiguities in English language. We believe that, by examining the specific language of a topic, we will help to identify how language and culture tools shape the understanding of students’ mathematical development. Although this study is based on Korean context, the method and findings will shed light on other East Asian languages, and will add value to the research on international comparative studies.

For a graph X with at most one isolated vertex and without isolated edges, a product-irregular labeling ω : E ( X ) → { 1 , 2 , … , s } , first defined by Anholcer in 2009, is a labeling of the edges of X such that for any two distinct vertices u and v of X the product of labels of the edges incident with u is different from the product of labels of the edges incident with v . The minimal s for which there exist a product irregular labeling is called the product irregularity strength of X and is denoted by p s ( X ) . In this paper it is proved that p s ( X ) ≤ | V ( X ) | - 1 for any graph X with more than 3 vertices. Moreover, the connection between the product irregularity strength and the multidimensional multiplication table problem is given, which is especially expressed in the case of the complete multipartite graphs.