Novel way to generate infinite number of solutions to ancient maths puzzle

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Washington, March 15 : Two experts claim that they have found solutions to a mathematical problem that has been around for centuries.

Mathematician Daniel J. Madden and retired physicist Lee W. Jacobi say that they have discovered a way to generate an infinite number of solutions for a puzzle called 'Euler's Equation of degree four'.

The equation belongs to a branch of mathematics called number theory, which deals with the properties of numbers and the way they relate to each other, and 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.

An article describing the two expert's work, published in the journal The American Mathematical Monthly, says that mathematical equations are puzzles that need certain solutions "plugged into them" in order to create a statement that obeys the rules of logic.

This statement has been described with the help of an example of the equation x + 2 = 4, which does not work when x is replaced by "3" instead of "2".

The problem on which Jacobi and Madden were working was to satisfy a Diophantine equation, so named because they were first studied by the ancient Greek mathematician Diophantus, known as 'the father of algebra'.

They were finding which numeric substitutes for a, b, c and d could satisfy the equation a4 + b4 +c4 +d4 = (a + b + c + d)4.

All the solutions they have found so far are very large numbers.

Madden and Jacobi used elliptic curves to generate new solutions. They say that each solution contains a seed for creating more solutions, which is much more efficient than previous methods used.

So far, people have used computers to find new solutions, which requires a lot of computing time and power to analyse huge amounts of data.

The two experts say that with their method, people can now as many solutions as they wish. They say that there are an infinite number of solutions to this problem, and they have found a way to find them all.


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