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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 work describes the contributions of Leonhard Euler (1707-1783), the 18th- century physicist and mathematician, to the fields of articulatory and experimental phonetics. First the authors provide evidence for Euler's role in establishing the St. Petersburg Academy prize of 1780. Next, they consider a short and posthumously published work of Euler, the Meditatio de formatione vocum. It is shown that the Meditatio represents an early attempt to compare vowels in several languages, and includes a two-dimensional classification of vowels which anticipates in many ways the International Phonetic Alphabet vowel chart. [ABSTRACT FROM AUTHOR]

It is fairly well known that Darwin was inspired to formulate his theory of natural selection by reading Thomas Malthus's Essay on the Principle of Population. In fact, by reading Darwin's notebooks, we can even locate one particular sentence which started Darwin thinking about population and selection. What has not been done before is to explain exactly where this sentence - essentially Malthus's ideas about geometric population growth - came from. In this essay we show that eighteenth century mathematician Leonhard Euler is responsible for this sentence, and in fact forms the beginning of the logical chain which leads to the creation of the theory of natural selection. We shall examine the fascinating path taken by a mathematical calculation, the many different lenses through which it was viewed, and the path through which it eventually influenced Darwin. [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.