Noam Elkies: Summing the Fourth Powers

A few weekends ago, I went to the MOVES (Mathematics of Various Entertaining Subjects) conference held by the Museum of Mathematics in New York City. I gave a workshop on the Dynamic World of Mathematics, and got to see some very interesting presentations and people.

Meeting Noam Elkies at MOVES

One of the people who I met was Noam Elkies. Noam Elkies is currently a professor at Harvard, and is the youngest full tenured professor in the history of the school.

In 1772, Leonhard Euler conjectured that there was no way to solve the following equation with whole numbers:

a⁴ = b⁴ + c⁴ + d⁴

For over 200 years, mathematicians were trying to prove this conjecture, but to no avail. Finally, in 1986, a twenty-year-old Noam Elkies offered a different type of answer.

In most posts, I will explain how someone proved a conjecture true. But in this instance, Noam Elkies proved the conjecture false. He found values of a, b, c, and d that made the equation true, and thereby showing that the equation is possible.

a = 20,615,673
b = 18,796,760
c = 15,365,649
d = 2,682,440

In addition, he proved that there are an infinite number of solutions to this equation. That completely went against what Euler had thought! This shows that you can never be 100% sure if something is true until you have proof. That philosophy applies to math, science, critical thinking in general, and it is also a great story to show the importance of mathematical proofs.

Ironically, my TEDx talk from this last June just went up a week and a half ago, and I told that exact story in my talk. So, I thought this would be an appropriate post to share the video in.