Arithmetica

"Arithmetica" is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century CE. It is a collection of 130 algebra problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations.

Equations in the book are called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the "Arithmetica" problems lead to quadratic equations. It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, which states that for the equation $x^n+y^n=z^n$ where x, y, and z are integers not equal to zero, n cannot be an integer greater than 2.

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form (4n + 3) cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler.

"Arithmetica" became known to the Arabs sometime before the tenth century [cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=234|isbn=0471543977|quote=Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine "Arithmetica" became familiar before the end of the tenth century.] when Abu'l-Wefa translated it into Arabic. [cite book|first=Carl B.|last=Boyer|authorlink=Carl Benjamin Boyer|title=A History of Mathematics|edition=Second Edition|publisher=John Wiley & Sons, Inc.|year=1991|chapter=The Arabic Hegemony|pages=239|isbn=0471543977|quote=Abu'l-Wefa was a capable algebraist as well as a trigonometer. He commented on al-Khwarizmi's "Algebra" and translated from Greek one of the last great classics - The "Arithmetica" of Diophantus.]

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