Mathematicians find new solutions to an ancient puzzle

Many people find complex math puzzling, including some mathematicians.

Recently, mathematician Daniel J. Madden and retired physicist, Lee W. Jacobi, found solutions to a puzzle that has been around for centuries.

Jacobi and Madden have found a way to generate an infinite number of solutions for a puzzle known as 'Euler’s Equation of degree four.'

The equation is part of a branch of mathematics called number theory. Number theory deals with the properties of numbers and the way they relate to each other. It is filled with problems that can be likened to numerical puzzles.

“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, an associate professor of mathematics at The University of Arizona in Tucson.

The team's finding is published in the March issue of The American Mathematical Monthly.

Equations are puzzles that need certain solutions “plugged into them” in order to create a statement that obeys the rules of logic.

For example, think of the equation x + 2 = 4. Plugging “3” into the equation doesn’t work, but if x = 2, then the equation is correct.

In the mathematical puzzle that Jacobi and Madden worked on, the problem was finding variables that satisfy a Diophantine equation of order four. These equations are so named because they were first studied by the ancient Greek mathematician Diophantus, known as 'the father of algebra.’

In its most simple version, the puzzle they were trying to solve is the equation: (a)(to the fourth power) + (b)(to the fourth power) + (c)(to the fourth power) + (d)(to the fourth power) = (a + b + c + d)(to the fourth power)

That equation, expressed mathematically, is: a⁴ + b⁴ +c⁴ +d⁴ = (a + b + c + d)⁴

Madden and Jacobi found a way to find the numbers to substitute, or plug in, for the a's, b's, c's and d's in the equation. All the solutions they have found so far are very large numbers.

In 1772, Euler, one of the greatest mathematicians of all time, hypothesized 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.

Euler's hypothesis was disproved in 1987 by a Harvard graduate student named Noam Elkies. He found a case where only three variables were needed. Elkies solved the equation: (a)(to the fourth power) + (b)(to the fourth power) + (c)(to the fourth power) = e(to the fourth power), which shows only three variables are needed to create a variable that is a fourth power.