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The Devil’s Advocate and the binomial expansion 魔鬼代言人与二项式展开
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.36
Nick Lord
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引用次数: 0
Infinitely many composites 无限多的复合材料
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.4
Nick Lord, Des MacHale
{"title":"Infinitely many composites","authors":"Nick Lord, Des MacHale","doi":"10.1017/mag.2024.4","DOIUrl":"https://doi.org/10.1017/mag.2024.4","url":null,"abstract":"In number theory, we frequently ask if there are infinitely many prime numbers of a certain type. For example, if n is a natural number:\u0000(i)Are there infinitely many (Mersenne) primes of the form 2n − 1?(ii)Are there infinitely many primes of the form n2 + 1?These problems are often very difficult and many remain unsolved to this day, despite the efforts of many great mathematicians. However, we can sometimes comfort ourselves by asking if there are infinitely many composite numbers of a certain type. These questions are often (but not always) easier to answer. For example, echoing (i) above, we can ask if there are infinitely many composites of the form 2p − 1 with p a prime number but (to the best of our knowledge) this remains an unsolved problem. Of course, it must be the case that there are either infinitely many primes or infinitely many composites of the form 2p − 1 and it seems strange that we currently cannot decide on either of them.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"177 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139836048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
108.11 Euler’s limit—revisited 108.11 欧拉极限重温
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.28
Bikash Chakraborty, Sagar Chakraborty
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引用次数: 0
A cautionary tale about the pole of polar coordinates 极坐标极点的警示故事
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.37
Nick Lord
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引用次数: 0
xy = cos (x + y) and other implicit equations that are surprisingly easy to plot xy = cos (x + y) 及其他隐式方程，绘制起来出奇地容易
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.2
Michael Jewess
{"title":"xy = cos (x + y) and other implicit equations that are surprisingly easy to plot","authors":"Michael Jewess","doi":"10.1017/mag.2024.2","DOIUrl":"https://doi.org/10.1017/mag.2024.2","url":null,"abstract":"The following equations relate y only implicitly to x:(1)(2) In both equations, y is a function of x for a continuous range of (x, y) values in the real x-y plane. (1) represents an ellipse. (2) has been designed by the author to have a solution in the real x-y plane at (−1, 2), and because the function on the left-hand side of (2) meets certain conditions regarding continuity and partial differentiability there must be a line of points in the real x-y plane satisfying (2) and passing continuously through (−1, 2) [1, pp. 23-28].","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"117 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139835957","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Singular matrices and pairwise-tangent circles 奇异矩阵和对切圆
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.3
A. Beardon
{"title":"Singular matrices and pairwise-tangent circles","authors":"A. Beardon","doi":"10.1017/mag.2024.3","DOIUrl":"https://doi.org/10.1017/mag.2024.3","url":null,"abstract":"The idea of using the generalised inverse of a singular matrix A to solve the matrix equation Ax = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette. Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence (v1, …, vn) of its distinct, consecutive, vertices. It is convenient to let vn + 1 = v1 and (later) Cn + 1 = C1.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"11 ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139836359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lost in the Math Museum by Colin Adams, pp 209, $35 (paperback), ISBN 978-1-47046-858-3, American Mathematical Society (2022) 迷失在数学博物馆》，科林-亚当斯著，第 209 页，35 美元（平装本），国际标准书号 978-1-47046-858-3，美国数学学会（2022 年）。
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.53
Mark Hunacek
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引用次数: 0
The role of convexity in defining regular polyhedra 凸性在定义正多面体中的作用
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.10
Chris Ottewill
{"title":"The role of convexity in defining regular polyhedra","authors":"Chris Ottewill","doi":"10.1017/mag.2024.10","DOIUrl":"https://doi.org/10.1017/mag.2024.10","url":null,"abstract":"It is almost twenty years since Branko Grünbaum lamented that the ‘original sin’ in the theory of polyhedra is that from Euclid onwards “the writers failed to define what are the ‘polyhedra’ among which they are finding the ‘regular’ ones” ([1, p. 43]). Various definitions of ‘regular’ can be found in the literature with a condition of convexity often included (e.g. [2, p. 301], [3, p. 77], [4, p. 47], [5, p. 435], [6, p. 16]). The condition of convexity is usually cited to exclude regular self-intersecting polyhedra, i.e. the Kepler-Poinsot polyhedra, such as the ‘great dodecahedron’ consisting of twelve intersecting pentagonal faces shown in Figure 1 with one face shaded. Richeson also notes ([4, pp. 47-48]) that, for a particular definition of ‘regular’, convexity is needed to exclude the ‘punched-in’ icosahedron shown in Figure 2.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"59 20","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139775034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Relating constructions and properties through duality 通过二元性关联构造和属性
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.5
Steven J. Kilner, David L. Farnsworth
{"title":"Relating constructions and properties through duality","authors":"Steven J. Kilner, David L. Farnsworth","doi":"10.1017/mag.2024.5","DOIUrl":"https://doi.org/10.1017/mag.2024.5","url":null,"abstract":"Our goal is to find new constructions and properties of parabolas. Our strategy is to display the steps in a known construction or property, and then to take the dual of the steps in order to create a new construction or property.","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"12 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139776213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
108.01 A use of Pythagorean triples in a problem in elementary geometry 108.01 勾股定理三段论在初等几何问题中的应用
The Mathematical Gazette Pub Date : 2024-02-15 DOI: 10.1017/mag.2024.18
Alexander Kronberg
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引用次数: 0
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