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: (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.
{"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}
{"title":"A cautionary tale about the pole of polar coordinates","authors":"Nick Lord","doi":"10.1017/mag.2024.37","DOIUrl":"https://doi.org/10.1017/mag.2024.37","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"105 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139834828","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}
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].
{"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}
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.
{"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}
{"title":"Lost in the Math Museum by Colin Adams, pp 209, $35 (paperback), ISBN 978-1-47046-858-3, American Mathematical Society (2022)","authors":"Mark Hunacek","doi":"10.1017/mag.2024.53","DOIUrl":"https://doi.org/10.1017/mag.2024.53","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"2 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139774634","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}
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.
自布兰科-格伦鲍姆（Branko Grünbaum）感叹多面体理论的 "原罪 "在于从欧几里得开始 "作者们未能定义什么是'多面体'，而他们要在其中找出'正则'多面体"（[1, 第 43 页]）以来，已经过去将近二十年了。文献中关于 "正则 "的定义多种多样，通常都包含凸性条件（例如 [2, p. 301]，[3, p. 77]，[4, p. 47]，[5, p. 435]，[6, p. 16]）。凸性条件通常被用来排除规则的自相交多面体，即开普勒-平素多面体，如图 1 所示由十二个相交的五边形面组成的 "大十二面体"，其中一个面是阴影。Richeson 还指出（[4, 第 47-48 页]），根据 "正则 "的特定定义，凸性是排除图 2 所示的 "打孔 "二十面体的必要条件。
{"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}
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.
{"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}
{"title":"108.01 A use of Pythagorean triples in a problem in elementary geometry","authors":"Alexander Kronberg","doi":"10.1017/mag.2024.18","DOIUrl":"https://doi.org/10.1017/mag.2024.18","url":null,"abstract":"","PeriodicalId":22812,"journal":{"name":"The Mathematical Gazette","volume":"144 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139834783","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}
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