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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]

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 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.