Euler’s Smallest Squares

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Subject
Euler, number theoryAbstract
Leonhard Euler approached a problem involving x, y, and z for which x^2+y^2+z^2 and x^2 y^2+y^2 y^2+x^2 z^2 are perfect squares. Euler believed that he had found the absolute smallest solution to this problem. However, Euler made mathematical mistakes when he was calculating the formulas he used to derive the trios. Thus, depending on the specified definition of the word small, smaller solutions that satisfy the original problem were discovered. We will find that some definitions of small do in fact have smaller solutions then the ones that Euler found, and that Euler’s methods when approaching this problem may not be the most efficient.