The great mathematician Leonard Euler (1707 – 1783) was comfortable working in any branch of mathematics (and he even invented some new branches, such as topology). This post is about one of his many results in number theory. I will describe a formula that he discovered in 1737 which involves fractions formed from prime numbers. The formula is an odd and surprising result, and its derivation illustrates Euler’s remarkable ingenuity.

For more than one hundred years, this formula was just another curiosity among Euler’s many results. Then, in 1859 Bernard Reimann used it as the starting point in his landmark paper on prime numbers.

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When math textbooks need an example of a right triangle, they frequently use a triangle with sides of length 3, 4, and 5, since the numbers work out so nicely: \(3^{2}+4^{2}=5^{2}\) by the Pythagorean theorem. If that gets tiresome, 12, 5, 13 might be used: \(5^{2}+12^{2}=13^{2}\). Clearly, multiplies of these numbers work also, e.g. \(6^{2}+8^{2}=10^{2}\).
Such [...]

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