Lee Jacobi enjoys playing with numbers

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A retired Tucson physicist and a UA mathematician have solved a problem in mathematics that had gone unanswered for more than 200 years.

In 1772, Leonhard Euler stated that in equations with an exponent greater than one, the number of variables would have to equal the exponent.

Of course, the famous Pythagorean theorem, a2+b2 = c2 was proved, but Euler, one of the greatest mathematicians ever, theorized that this would apply to other exponents as well.

So, a3+b3+c3 would equal (a+b+c)3 and so on.

Flash-forward to 1987 and Euler, despite his brilliance, is proved wrong by Noam Elkies, at the time a graduate student at Harvard University. Elkies discovered an equation, a4+b4+c4 = d4, bypassing Euler's theorized need for another variable and making his own name in mathematics.

Elkies' work excited a lot of interest in number theory among amateur mathematicians. In 1989, Lee Jacobi, a physicist working in the defense industry, including at Raytheon Missile Systems, began "playing around" with number theory, including some of Euler's work. When he retired in 2003, Jacobi began to work more extensively on the problem and eventually contacted Daniel Madden, an associate professor of mathematics at the University of Arizona, to look at his work.

"He really had a couple of new ideas," Madden said.

Jacobi was attempting to show that Euler had been correct in thinking that the number of variables could equal the exponent in an equation with an exponent of four, and after a lot of work, he and Madden proved it mathematically.

Their paper was published in the March issue of the American Mathematical Monthly.

They "established the truth of this equation," Madden said by "bringing modern techniques to bear" on the problem.

"When it boils down to it, I used algebra," Jacobi said.

Jacobi had been working with specific numbers, but it was only with the help of Madden that the equation turned into something universal.

"All the big ideas were his," Madden said, adding that there are a lot of amateurs in mathematics who add to the growth of knowledge in the field.

To check their work, the duo used Mathematica, a computer program for math that can do a lot more work much faster than can be done by hand.

Euler "would have died to have such a thing," Jacobi said.

The new work is "like a cousin to that (Elkies') equation," Jacobi said, related, but still different.

Although this kind of math might seem incredibly esoteric and remote from average life, Madden said it really is just an extension of work that carpenters and landscapers use all the time to make right angles, except with a fourth power instead of only a second power.

That's not to say the work is easy. The numbers used were so large that even the smallest among them could not be checked on a scientific calculator, Madden said.

Now though, instead of consuming a great deal of computer power to check these kinds of equations, the equation Madden and Jacobi published can be used, saving time and energy.

This will be useful to physicists, who use these types of equations frequently, such as in string theory work.

Jacobi, now 68, said he does not plan to stop playing around with numbers and number theory, though he said he will still run everything by Madden and other professionals.

"I'm the amateur in this," he said.