[a] Jacobi and Madden showed that there are an infinitude of solutions of this equation with all variables non-zero.
The Jacobi–Madden equation represents a particular case of the equation first proposed in 1772 by Leonhard Euler who conjectured that four is the minimum number (greater than one) of fourth powers of non-zero integers that can sum up to another fourth power.
Noam Elkies was first to find an infinite series of solutions to Euler's equation with exactly one variable equal to zero, thus disproving Euler's sum of powers conjecture for the fourth power.
[3] However, until Jacobi and Madden's publication, it was not known whether there exist infinitely many solutions to Euler's equation with all variables non-zero.
Jacobi and Madden noticed that a different starting value, such as found by Jaroslaw Wroblewski,[5] would result in a different infinite series of solutions.