There Are Infinite Solutions to Euler’s Equation of Degree Four

March 20, 2008

Mathematician Daniel J. Madden (University of Arizona) and retired physicist Lee W. Jacobi have found a way to generate an infinite number of solutions to Euler’s "Equation of degree four." All the solutions are very large numbers.

The problem, posed by Euler in 1772, was finding variables that satisfy the Diophantine equationa⁴ + b⁴ +c⁴ +d⁴ = (a + b + c + d)⁴.

“It’s like a puzzle: can you find four fourth powers that add up to another fourth power? Trying to answer that question is difficult because it is highly unlikely that someone would sit down and accidentally stumble upon something like that,” said Madden.

Madden and Jacobi used elliptic curves to generate solutions. Each solution contains a seed for creating more solutions. This approach is much more efficient than previous methods.

“Modern number theory allowed me to see with more clarity the implications of his (Jacobi’s) calculations,” Madden said. “It was a nice collaboration,” said Jacobi. “I have learned a certain amount of new things about number theory; how to think in terms of number theory, although sometimes I can be stubbornly algebraic.”

Their paper “On a⁴ + b⁴ +c⁴ +d⁴ = (a + b + c + d)⁴" was published in the March issue of The American Mathematical Monthly.