Pub Date : 2023-10-30DOI: 10.1080/0025570x.2023.2266348
Roger B. Nelsen
SummaryWe show visually that if p and q=2p+1 are each prime, then the product pq is 1 more than a multiple of 18. AcknowledgmentThe author wishes to thank an anonymous reviewer for helpful comments and suggestions on an earlier draft of this note.Additional informationNotes on contributorsRoger B. NelsenROGER NELSEN (MR Author ID: 237909) is a professor emeritus at Lewis & Clark College, where he taught mathematics and statistics for 40 years.
{"title":"Sophie Germain and Safe Prime Products Modulo 18","authors":"Roger B. Nelsen","doi":"10.1080/0025570x.2023.2266348","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266348","url":null,"abstract":"SummaryWe show visually that if p and q=2p+1 are each prime, then the product pq is 1 more than a multiple of 18. AcknowledgmentThe author wishes to thank an anonymous reviewer for helpful comments and suggestions on an earlier draft of this note.Additional informationNotes on contributorsRoger B. NelsenROGER NELSEN (MR Author ID: 237909) is a professor emeritus at Lewis & Clark College, where he taught mathematics and statistics for 40 years.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136023242","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}
Pub Date : 2023-10-20DOI: 10.1080/0025570x.2023.2266332
Nam Gu Heo
SummaryEuler’s inequality in Euclidean geometry is a famous result relating the circumradius and inradius of a triangle. We provide a new proof of this result using elementary ideas about area and isosceles triangles.MSC: 51M04 Additional informationNotes on contributorsNam Gu HeoNAM GU HEO (MR Author ID: 1106949) studied mathematics education and received an Ed.D. from the Korea National University of Education. Currently, he is a professor at Sunchon National University, Korea.
{"title":"Another Proof of Euler’s Inequality","authors":"Nam Gu Heo","doi":"10.1080/0025570x.2023.2266332","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266332","url":null,"abstract":"SummaryEuler’s inequality in Euclidean geometry is a famous result relating the circumradius and inradius of a triangle. We provide a new proof of this result using elementary ideas about area and isosceles triangles.MSC: 51M04 Additional informationNotes on contributorsNam Gu HeoNAM GU HEO (MR Author ID: 1106949) studied mathematics education and received an Ed.D. from the Korea National University of Education. Currently, he is a professor at Sunchon National University, Korea.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135569732","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}
Pub Date : 2023-10-20DOI: 10.1080/0025570x.2023.2266347
Elias Abboud
SummaryIn this article we consider uniform tessellations of the hyperbolic plane. Two counts are performed. The first, considers the number of polygons in layers of the hyptagrid and is shown to be related to Fibonacci numbers. The second, considers the number of fishes in layers of a superimposed octagrid on Escher’s circle limit III. To excute these counts we solve two linear recurrence relations, homogeneous and non-homogeneous. The initial conditions are set up by performing tessellations using a software in hyperbolic geometry.MSC: 51M10 AcknowledgmentsThe author is indebted to the Editor and an anonymous referee for their valuable comments that substantially improved the exposition of the paper. Special thanks are also due to Douglas Dunham for the permission to use his recreation of Escher’s Circle Limit III.Notes1 Note that the online version of this article has color diagrams.Additional informationNotes on contributorsElias AbboudELIAS ABBOUD (MR Author ID: 249090) received his D.Sc from the Technion-Haifa, Israel. Since 1992, he has taught Mathematics at Beit Berl College. Between the years 2010–2017. he served as the Math Chair in the Arab Academic Institution within the Faculty of Education of Beit Berl College. Since 2001, he also works partially at the Academic Arab College of Education-Haifa.
{"title":"Counting Polygons and Fishes in Uniform Tessellations of the Hyperbolic Plane","authors":"Elias Abboud","doi":"10.1080/0025570x.2023.2266347","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2266347","url":null,"abstract":"SummaryIn this article we consider uniform tessellations of the hyperbolic plane. Two counts are performed. The first, considers the number of polygons in layers of the hyptagrid and is shown to be related to Fibonacci numbers. The second, considers the number of fishes in layers of a superimposed octagrid on Escher’s circle limit III. To excute these counts we solve two linear recurrence relations, homogeneous and non-homogeneous. The initial conditions are set up by performing tessellations using a software in hyperbolic geometry.MSC: 51M10 AcknowledgmentsThe author is indebted to the Editor and an anonymous referee for their valuable comments that substantially improved the exposition of the paper. Special thanks are also due to Douglas Dunham for the permission to use his recreation of Escher’s Circle Limit III.Notes1 Note that the online version of this article has color diagrams.Additional informationNotes on contributorsElias AbboudELIAS ABBOUD (MR Author ID: 249090) received his D.Sc from the Technion-Haifa, Israel. Since 1992, he has taught Mathematics at Beit Berl College. Between the years 2010–2017. he served as the Math Chair in the Arab Academic Institution within the Faculty of Education of Beit Berl College. Since 2001, he also works partially at the Academic Arab College of Education-Haifa.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135569727","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}
Pub Date : 2023-08-08DOI: 10.1080/0025570X.2023.2231840
Jingkai Zhou, Wenjie Zhou
Summary The 2019 Romanian Master of Mathematics Competition included an intriguing problem in graph theory. We present a concise solution that differs from that presented in the official competition documents.
{"title":"An Intriguing Graph Theory Problem From RMM 2019","authors":"Jingkai Zhou, Wenjie Zhou","doi":"10.1080/0025570X.2023.2231840","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231840","url":null,"abstract":"Summary The 2019 Romanian Master of Mathematics Competition included an intriguing problem in graph theory. We present a concise solution that differs from that presented in the official competition documents.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44927259","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}
Pub Date : 2023-08-08DOI: 10.1080/0025570x.2023.2234203
Jason Rosenhouse
{"title":"Letter from the Editor","authors":"Jason Rosenhouse","doi":"10.1080/0025570x.2023.2234203","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2234203","url":null,"abstract":"","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840649","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}
Pub Date : 2023-08-08DOI: 10.1080/0025570x.2023.2237380
{"title":"Problems and Solutions","authors":"","doi":"10.1080/0025570x.2023.2237380","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2237380","url":null,"abstract":"","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135840650","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}
Pub Date : 2023-07-26DOI: 10.1080/0025570X.2023.2234243
Benjamin W. L. Margolis
Summary We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.
{"title":"Another Geometric Interpretation of Cramer’s Rule","authors":"Benjamin W. L. Margolis","doi":"10.1080/0025570X.2023.2234243","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2234243","url":null,"abstract":"Summary We develop a geometric interpretation of Cramer’s rule as a generalization of projection onto orthogonal basis vectors using the rows of the adjugate. This interpretation makes connections between elementary linear algebra concepts like the solution to linear equations, inner products, and projections. Such connections are useful for introducing broader concepts related to Hilbert spaces and geometric algebras like Grassman algebra. Such connections were essential for the author’s mathematical education as an engineer.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41386521","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}
Pub Date : 2023-07-20DOI: 10.1080/0025570x.2023.2231808
R. Nelsen
Summary We show wordlessly that if n is a sum of four distinct integer squares, then so is n 2, in four different ways.
摘要我们用四种不同的方式无言地证明，如果n是四个不同整数平方的和，那么n2也是。
{"title":"Sums of Four Squares","authors":"R. Nelsen","doi":"10.1080/0025570x.2023.2231808","DOIUrl":"https://doi.org/10.1080/0025570x.2023.2231808","url":null,"abstract":"Summary We show wordlessly that if n is a sum of four distinct integer squares, then so is n 2, in four different ways.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45241153","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}
Pub Date : 2023-07-20DOI: 10.1080/0025570X.2023.2231830
Timothy E. Goldberg, L. Wilson
Summary A golden rectangle is characterized by the fact that if an inscribed square is removed from one end, then the remaining rectangle is similar to the original one. By iterating this process of removing a square, one obtains an infinite sequence of nested golden rectangles which converges to a point. One can construct other sequences of rectangles by starting from arbitrary, not necessarily golden, rectangles. The goal of this paper is to analyze the behavior of these sequences, primarily by modeling the process using linear algebra.
{"title":"Limits of Golden Constructions","authors":"Timothy E. Goldberg, L. Wilson","doi":"10.1080/0025570X.2023.2231830","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231830","url":null,"abstract":"Summary A golden rectangle is characterized by the fact that if an inscribed square is removed from one end, then the remaining rectangle is similar to the original one. By iterating this process of removing a square, one obtains an infinite sequence of nested golden rectangles which converges to a point. One can construct other sequences of rectangles by starting from arbitrary, not necessarily golden, rectangles. The goal of this paper is to analyze the behavior of these sequences, primarily by modeling the process using linear algebra.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41608417","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}
Pub Date : 2023-07-20DOI: 10.1080/0025570X.2023.2231335
Michael Z. Spivey
Summary The binomial coefficients can be used to count the number of ways selection can be made both with and without repetition. We give a combinatorial interpretation of the binomial coefficients that generalizes both of these cases to partial repetition.
{"title":"The Binomial Coefficients and Selection with Partial Repetition","authors":"Michael Z. Spivey","doi":"10.1080/0025570X.2023.2231335","DOIUrl":"https://doi.org/10.1080/0025570X.2023.2231335","url":null,"abstract":"Summary The binomial coefficients can be used to count the number of ways selection can be made both with and without repetition. We give a combinatorial interpretation of the binomial coefficients that generalizes both of these cases to partial repetition.","PeriodicalId":18344,"journal":{"name":"Mathematics Magazine","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42060020","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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