Lagrange and the four-square theorem - Archive ouverte HAL

In this article we will first provide an overview of Lagrange’s arithmetic work, written between 1768 and 1777. We then focus on the proof of the four-square theorem published in 1772, and shed light on the context of its implementation by relying on other memoirs by Lagrange and Leonhard Euler. By means of this analysis, our goal is to give a concrete vision of the arithmetic, algebraic and analytical methods and tools used by Lagrange in number theory and place his arithmetic practice in the context of the second half of the eighteenth century.

Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares

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Lagrange's four-square theorem - Wikipedia

Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares

Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares

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Pourchet's theorem in action: decomposing univariate nonnegative polynomials as sums of five squares