Jacobi–Madden equation

The Jacobi–Madden equation is a diophantine equation

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

proposed by the physicist Lee W. Jacobi and the mathematician Daniel J. Madden in 2008.[1][2]
History

The Jacobi–Madden equation represents a particular case of the equation

\[ a^4 + b^4 +c^4 +d^4 = e^4 \, \]

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. This conjecture, now known as Euler's sum of powers conjecture, was a natural generalization of the Fermat's Last Theorem that was proved for the fourth power by Pierre de Fermat himself.

Noam Elkies was first to find an infinite series of solutions to the Euler's equation with just one variable equal zero, thus disproving the 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 the Euler's equation with all variables non-zero. Only a finite number of such solutions was known.[4][5] One of these solutions, discovered by Simcha Brudno in 1964,[6] yielded a solution to the Jacobi–Madden equation:

\[ 5400^4 + 1770^4 + (-2634)^4 + 955^4 = (5400 + 1770 - 2634 + 955)^4. \, \]

Approach

Jacobi and Madden used Brudno's solution and a certain elliptic curve to construct an infinite series of solutions to both the Jacobi–Madden and Euler's equations. In contrast to Elkies' solutions, these new solutions expose non-zero values of the variables.

Jacobi and Madden also noticed that a different starting solution, such as

\[ (-31764)^4 + 27385^4 + 48150^4 + 7590^4 = (-31764 + 27385 + 48150 + 7590)^4 \, \]

found by Jaroslaw Wroblewski,[5] would result in a different infinite series of solutions.[7]

References

^ Jacobi, Lee W.; Madden, Daniel J. (2008). "On a^4 + b^4 + c^4 + d^4 = (a + b + c + d)^4". American Mathematical Monthly 115 (3): 220–236.
^ Mathematicians find new solutions to an ancient puzzle
^ Noam Elkies (1988). "On A4 + B4 + C4 = D4". Mathematics of Computation 51 (184): 825–835. doi:10.2307/2008781. MR0930224.
^ Weisstein, Eric W., "Diophantine Equation–4th Powers" from MathWorld.
^ a b Jaroslaw Wroblewski Database of solutions to the Euler's equation
^ Simcha Brudno (1964). "A further example of A4 + B4 + C4 + D4 = E4". Mathematical Proceedings of the Cambridge Philosophical Society 60 (04): 1027–1028. doi:10.1017/S0305004100038470. MR0166151.
^ Seiji Tomita Solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4