Everyone who knows algebra is familiar with the quadratic formula for solving second degree equations.
It has been known since at least the 9th century.

The logical next step in algebra was to find the corresponding formula for the 3rd degree (cubic) equation. Starting about 1520, a group of Italian mathematicians competed to find the fabled cubic formula, and in 1545, one of them, Girolamo Cardano (1501 – 1576), published the solution. As we will see, the formula is complex, and of limited practical use. However, the cubic problem has great historical importance. Not only was the art of algebra was advanced by the effort to find it, but mathematicians eventually realized that the formula would only worked correctly if they faced up to the existence of complex numbers.

In this post, I will show a derivation of the formula, then show how it is used.

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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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