Two years ago, a couple of high-school classmates each composed a mathematical marvel. It was a trigonometric proof of the Pythagorean theorem. Now, at age 19, they’ve just reported finding 10 more proofs of it.

For more than 2,000 years, such proofs were considered impossible. That didn’t deter Ne’Kiya Jackson and Calcea Johnson. The pair published their new ones October 28 in American Mathematical Monthly.

“Some people have the impression that you have to be in academia for years and years before you can actually produce some new mathematics,” says Álvaro Lozano-Robledo. A mathematician, he works at the University of Connecticut in Storrs. But Jackson and Johnson show that “you can make a splash even as a high-school student.”

Jackson is now studying pharmacy at Xavier University of Louisiana in New Orleans. Johnson is studying environmental engineering at Louisiana State University in Baton Rouge.

Math of triangles

Mathematical proofs are sequences of statements that demonstrate an assertion is true or false. The teens tackled Pythagoras’ theorem. This has to do with a right triangle. That’s one having a corner that’s a 90-degree angle (the same as each corner of a rectangle).

In math class, students learn a rule that lets you know the length of a right triangle’s two shorter sides. The length of each, squared (meaning each multiplied by itself), when added together equals the square of its hypotenuse (or longest side). The rule is written as a2 + b2 = c2. In this algebraic phrase, a and b are the right triangle’s shorter two sides. The third, c, is the hypotenuse.

This formula for the right triangle has been proven many times using algebra and geometry. But in 1927, mathematician Elisha Loomis made an assertion. It could never be proved, he said, using the rules of trigonometry (Trig-uh-NOM-eh-tree). Often just called “trig,” this type of geometry — or study of shapes — deals with the relationships between angles and the lengths of triangles’ sides.

Loomis argued that any trig-based proof of the Pythagorean theorem would fail. Why? It would use circular logic, a type of false reasoning that only works by first assuming its conclusion is true. Many trigonometry rules assume the Pythagorean theorem is true. So trying to prove Pythagoras’ theorem using trig would rely on rules based on the theorem, Loomis argued. In other words, one would have to assume that the theorem was true before proving it was true.

Simplifying your point of view

Mathematician Jason Zimba announced the first trig-based proof of Pythagoras’ theorem in 2009. The second came six years later, from mathematician Nuno Luzia. These were the only two until 2022, when Jackson and Johnson presented their first proofs. At the time, both were seniors at St. Mary’s Academy. It’s a private school for young Black women in New Orleans.

The teens first formally presented their work at an American Mathematical Society meeting in March 2023. Afterward, the duo set out to publish their findings in a peer-reviewed math journal.

“This proved to be the most daunting task of all,” they noted in their paper. They had to write up their work, plus learn new skills, like coding. At the same time, both were starting college.

But working on their early proof and studying Zimba’s “sparked the creative process,” Jackson says. “From there we developed additional proofs.”

And, Johnson adds, “It was important to me to have our proofs published to solidify that our work is correct and respectable.”

Trigonometric terms can be defined in two different ways, the young women note. And that can complicate efforts to prove Pythagoras’ theorem. So they focused on just one of these definitions. From that, they developed four proofs for right triangles with sides of different lengths. They also created one for right triangles with two equal sides.

This short video shows how to use the Pythagorean theorem to find out if a three-sided polygon is a “right” triangle, meaning one having a 90-degree angle.

Among these, one proof stands out to Lozano-Robledo. In it, the students fill one larger triangle with an infinite sequence of smaller triangles. Then they use calculus (another type of math) to find the lengths of the larger triangle’s sides. “It looks like nothing I’ve ever seen,” Lozano-Robledo says.

Jackson and Johnson left another five proofs “for the interested reader to discover,” they wrote. The paper provides a lemma — a sort of stepping-stone to proving a theorem. That “provides a clear direction towards the additional proofs,” Johnson says.

Publishing the proofs was a big achievement. But that doesn’t mean the story is over.

Now, “other people might take the paper and generalize those proofs, or generalize their ideas, or use their ideas in other ways,” Lozano-Robledo says. “It just opens up a lot of mathematical conversations.”

Jackson hopes that the proofs will inspire other students to “see that obstacles are part of the process. Stick with it, and you might find yourself achieving more than you thought possible.”