Euler's equation of degree four

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Euler's equation of degree four is a mathematical problem proposed by Leonhard Euler in 1772.^[1] The problem, which deals with number theory, asks for a solution to the equation

$a^4 + b^4 +c^4 +d^4 = (a + b + c + d)^4,\,$

and had remained largely unsolved until early 2008, when mathematician Daniel J. Madden and physicist Lee W. Jacobi used elliptic curves to solve it, resulting in a proof that yields an infinite number of solutions to the equation. Until the breakthrough, 88 other solutions had been found, though it had not been proven if there were an infinite number of them. Madden and Jacobi's solution is somewhat recursive in that each solution contains a seed for another solution.^[2]

This puzzle was part of Euler's hypothesis that to satisfy equations with higher powers, there would need to be as many variables as that power. For example, a fourth order equation would need four different variables, like the equation above. This hypothesis was disproved in 1987 by a Harvard graduate student named Noam Elkies.