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In 1749, Leonhard Euler solved a longstanding open problem in number theory, proving Fermat's 1640 conjecture that any prime number of the form 4 m + 1 can be written as the sum of two squares. This achievement overshadowed an equally impressive calculation that Euler devised at the same time, to test numbers for primality. In this article, we explore Euler's primality test, demonstrating its computational parity with trial division, at least for a class of testable numbers. Additionally, we will see that primality testing was one of Euler's lifelong interests, and was a topic he returned to time and again as an application of his number theory work. [ABSTRACT FROM AUTHOR]

This study investigates changes in instructor teaching tendencies, instructor's perception of impact on student learning and dispositions, and methods of implementation of Primary Source Projects (PSPs). PSPs are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. In essence, they are a form of inquiry-based learning that incorporates the history of mathematics through original source texts. We provide an overview of results from two semesters of implementation reports and surveys administered at the beginning and end of the semester by instructors who implemented PSPs in their undergraduate mathematics classes. [ABSTRACT FROM AUTHOR]

This paper describes efforts by the author to engage and motivate students in undergraduate research by giving them a large audience and engaging the media in disseminating their work. I provide an introduction to public relations from the point of view of a mathematics professor, and describe some lessons I have learned in my own attempts to engage newsprint, radio, and television sources in stories about undergraduate research in mathematics. After describing some partially successful early attempts, the paper discusses a recent event in which undergraduate research in mathematics became, briefly, a “hot news item” in Central Washington. The paper concludes with some thoughts about the benefits of this type of work to our students and our profession. [ABSTRACT FROM AUTHOR]

The article discusses the study conducted to examine the reciprocal sum as a knowledge metric and also discusses its mathematical theory, computation and perfect numbers. Topics discussed include long and intimate connection of reciprocal sums with the number theory, using Euler's theorem for getting the better results of sum of reciprocals, and using partial summation as a powerful and versatile method for estimating the reciprocal sums.

The article analyzes the distribution of average playoffs found by playing a fixed number of St. Petersburg games. It examines the most popular experiment on the said game which is the results of the Compte de Buffon's playing the game 2048 times. It mentions that the St. Petersburg game is a probabilistic thought investigation which appears to have an infinite expected value, but which no person could be expected to pay much to play.

Editor's Note: As of this issue, The College Mathematics Journal has published fifty completed volumes. To celebrate 50 years of continued excellence in publishing expository mathematics, pedagogical discussions, and more, we plan to use two pages in each of the five issues of 2020 to tell the story of one decade of the journal. Some of these pieces will be written by people who were personally involved with CMJ during that time, so watch this space for some familiar names! [ABSTRACT FROM AUTHOR]

The article presents information on the properties of Graeco-Latin squares enumerated by mathematician Leonhard Euler. Euler suggested that a Graeco-Latin square of size n could never exist for any n of the form 4k +2, although he was not able to prove it. A Latin square is an n-by-n array of n distinct symbols in which each symbol appears exactly once in each row and column. On the other hand a Graeco-Latin square is an n-by-n array of ordered pairs from a set of n symbols such that in each row and each column of the array, each symbol appears exactly once in each coordinate. In one of his papers on Graeco-Latin squares, Euler used magic squares, which are closely related to Graeco-Latin squares. Magic squares were constructed by using Graeco-Latin squares of orders 3, 4 and 5. He showed that a Graeco-Latin square of order n can be converted into a magic square by the use of an algorithm. One can construct Graeco-Latin squares of every order n except those values for which the prime factorization of n contains only a single factor of 2.

A Latin square is a grid or array containing the same number of rows and columns. The cell entries are a sequence of symbols inserted in such a way that each symbol occurs only once in each row and only once in each column. Since the beginning of the twentieth century, Latin squares have been studied in the field of combinatorics. This article is concerned with the issue of who, according to the published literature, introduced Western mathematicians to Latin squares. The first mathematician to provide a depiction of Latin squares seems to have been Jacques Ozanam (1640–1718), who demonstrated the use of Latin squares in a problem based on playing cards. However, he did not discuss the analytic properties or the potential applications of Latin squares. The first mathematician to provide an analytic description of Latin squares seems to have been Leonhard Euler (1707–1783). [ABSTRACT FROM AUTHOR]