# Mathematicians find new solutions to an ancient puzzle

##### March 14, 2008

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.

Inspired by the accomplishments of the 22-year-old graduate student, Jacobi began working on mathematics as a hobby after he retired from the defense industry in 1989.

Fortunately, this was not the first time he had dealt with Diophantine equations. He was familiar with them because they are commonly used in physics for calculations relating to string theory.

Jacobi started searching for new solutions to the puzzle using methods he found in some number theory texts and academic papers.

He used those resources and Mathematica, a computer program used for mathematical manipulations.

Jacobi initially found a solution for which each of the variables was 200 digits long. This solution was different from the other 88 previously known solutions to this puzzle, so he knew he had found something important.

Jacobi then showed the results to Madden. But Jacobi initially miscopied a variable from his Mathematica computer program, and so the results he showed Madden were incorrect.

“The solution was wrong, but in an interesting way. It was close enough to make me want to see where the error occurred,” Madden said.

When they discovered that the solution was invalid only because of Jacobi’s transcription error, they began collaborating to find more solutions.

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

In the past, people found new solutions by using computers to analyze huge amounts of data. That required a lot of computing time and power as the magnitude of the numbers soared.

Now people can generate as many solutions as they wish. There are an infinite number of solutions to this problem, and Madden and Jacobi have found a way to find them all.

The title of their paper is, “On a4 + b4 +c4 +d4 = (a + b + c + d)4."

“Modern number theory allowed me to see with more clarity the implications of his (Jacobi’s) calculations,” Madden said.

“It was a nice collaboration,” Jacobi said. “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.”

Source: University of Arizona

Explore further: Super stair-climbers

## Related Stories

#### Super stair-climbers

August 19, 2016

The Balgrist campus is bustling with activity. After ten months and countless technical changes, the revised wheelchair from the Scewo team is ready for its first test drive. With one eye on the regulations and another on ...

#### World's fastest multiframe digital X-ray camera created

June 2, 2016

An adversary who steps inside a boxer's sense of rhythm may land a punch the boxer never saw coming.

#### Denmark champions wind power, sets record

January 10, 2015

Denmark has had a record year for wind power production. Denmark got 39.1 percent of its overall electricity from wind in 2014. That figure is according to the country's Climate and Energy Ministry.

#### CFOs indicate they would be willing to accept a solution to the 'fiscal cliff' that includes tax increases, survey finds

December 13, 2012

Chief financial officers say going over the "fiscal cliff" will lead to dramatic slowdowns in hiring and business spending in 2013 and will continue to hurt firms for years to come.

#### Three-year battery life for wireless human interface devices with new ultra-low-power 2.4-GHz wirelessUSB NX transceiver

May 15, 2014

Cypress Semiconductor Corp. today introduced its fourth-generation 2.4-GHz WirelessUSB radio-on-a-chip. The new WirelessUSB NX transceiver delivers Cypress's hallmark robust performance along with ultra-low power consumption, ...

#### Infineon Introduces World's Smallest HSPA+ Solution for 3G Smart Phones

February 15, 2010

Infineon Technologies today at the Mobile World Congress 2010 announced the availability of XMM 6260, the latest platform in its 3G slim-modem family.

## Recommended for you

#### New species of pterosaur discovered in Patagonia

August 30, 2016

Scientists today announced the discovery of a new species of pterosaur from the Patagonia region of South America. The cranial remains were in an excellent state of preservation and belonged to a new species of pterosaur ...

#### Statistical study offers evidence of warning signs before Neolithic community collapse

August 30, 2016

(Phys.org)—A trio of researchers, two with the University of Maryland and the other with University College London has found that early Neolithic communities exhibited warning signs before collapsing. In their paper published ...

#### Did fall from tree kill famous human ancestor Lucy?

August 29, 2016

The famous human ancestor known as Lucy walked the Earth, but it was her tree climbing that might have led to her demise, a new study suggests.

#### Ancient dental plaque sheds new light on the diet of Mesolithic foragers in the Balkans

August 29, 2016

The study of dental calculus from Late Mesolithic individuals from the site of Vlasac in the Danube Gorges of the central Balkans has provided direct evidence that Mesolithic foragers of this region consumed domestic cereals ...

#### Isotope study determines fish were more prominent in early Alaskans' diets

August 29, 2016

Ice age inhabitants of Interior Alaska relied more heavily on salmon and freshwater fish in their diets than previously thought, according to a newly published study.

#### Researchers plumb the secrets of tissue paper

August 24, 2016

Canada's tissue manufacturers are now much closer to producing the perfect paper, thanks to new UBC research.

##### Argiod
1 / 5 (2) Mar 15, 2008
I love it when a solution to a math problem turns out to be this elegant.
##### Doug_Huffman
1 / 5 (1) Mar 15, 2008
Would that its explanation here be as elegant and rise above 'equations as puzzles'
##### quantum_flux
not rated yet Mar 15, 2008
Whatever it is, I'll bet it has to do with 4 dimensional geometries with 3 spacial dimensions and 1 time dimension.

Kind of like the "A^2 plus B^2=(C)^2" rule for right triangles in plane geometry, but for the special case where "C = A plus B".
##### RAL
not rated yet Mar 16, 2008
I love it that an error in transcribing a number was turned into a whole new approach to the math.
##### koalabear3000
1 / 5 (1) Apr 16, 2008
i love it when i have dorks like u guys do my math homework for me. :]