Euler's equation of degree four

Euler's equation of degree four is a mathematical problem proposed by Leonhard Euler in 1772. [ [http://www.upi.com/NewsTrack/Science/2008/03/19/eulers_equation_of_degree_four_solved/8804/ 'Euler's equation of degree four' solved - UPI.com ] ] The problem, which deals with number theory, asks for a solution to the equation

: $a^4\; +\; b^4\; +c^4\; +d^4\; =\; (a\; +\; b\; +\; c\; +\; d)^4,,$ where { a, b, c, d } can be positive, negative, or zero integers. (See more below.)

This problem had remained largely unsolved until early 2008, when the mathematician Daniel J. Madden and the 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. [ [http://www.physorg.com/news124726812.html Mathematicians find new solutions to an ancient puzzle ] ]

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 the Harvard graduate student, Noam Elkies.

Restrictions and Possibilities on the Numbers

Obviously, the equation is true if a = b = c = d = 0. Also, in the set { a, b, c, d }, if three of the four are zero, then the equation is true.If two of the four are zero, then we get $a^4\; +\; b^4\; =\; (a\; +\; b)^4$, and Pierre de Fermat, himself, showed that this one is impossible for all nonzero numbers { a, b, c }, with $a^4\; +\; b^4\; =\; c^4$.

If one of the numbers is zero, then we get $a^4\; +\; b^4\; +\; c^4\; =\; (a\; +\; b\; +\; c)^4$. This is obviously not true for positive integers, because the right-hand side of the equation would be too large. Possibly,it might be true if one of them is allowed to be negative.

If none of the numbers is zero, then we get $a^4\; +\; b^4\; +\; c^4\; +\; d^4\; =\; (a\; +\; b\; +\; c\; +\; d)^4$. This is also obviously not true for positive integers, because the right-hand side of the equation would be too large. It has now been shown that there are infinitely-many solutions if some of the numbers are allowed to be negative, and some positive.

ee also

*Euler's sum of powers conjecture

References