Two teenagers once again proved an ancient mathematical rule

Two years ago, a pair of high school classmates created a mathematical miracle, a trigonometric proof of the Pythagorean theorem. They are now releasing 10 more.

For more than 2,000 years, such evidence was considered impossible. Still, undeterred, Ne’Kiah Jackson and Kalsea Johnson posted their new evidence October 28 in American Mathematical Monthly.

“Some people are under the impression that you have to spend years in academia before you can create new mathematics,” says mathematician Alvaro Lozano-Robledo of the University of Connecticut in Storr. But, he said, Jackson and Johnson show that “you can make a splash even as a high school student.”

Jackson is currently a pharmacy student at Louisiana Xavier University in New Orleans, and Johnson is studying environmental engineering at Louisiana State University in Baton Rouge.

Mathematical proofs are sequences of statements that demonstrate the truth or falsity of a statement. Pythagorean theorem — a² + b² = c²as regards the length of the hypotenuse of a right triangle to the length of its other two sides — has been proved many times by algebra and geometry (SN: 02.04.03).

But in 1927, mathematician Elisha Loomis declared that this feat could not be accomplished using the rules of trigonometry, the subset of geometry that deals with the relationships between the angles and side lengths of triangles. He believed that the Pythagorean theorem was so fundamental to trigonometry that any attempt to prove the theorem based on trigonometry would have to first assume that it was true, thus resorting to circular logic.

Jackson and Johnson proposed the first proofs based on trigonometry in 2022 as seniors at St. Mary’s Academy in New Orleans, a Catholic school attended mostly by young black women. At the time, there were only two other trigonometric proofs of the Pythagorean theorem, presented by mathematicians Jason Zimba and Nuno Luzia in 2009 and 2015, respectively. As Jackson says, working on the early proofs “spurred the creative process, and from there we developed additional proofs.”

After formally presenting his work at the March 2023 meeting of the American Mathematical Society, they decided to publish their findings in a peer-reviewed journal. “This turned out to be the most difficult task of all,” they said in the paper. In addition to writing, the duo had to develop new skills while attending college. “Learning to code in LaTeX (a typesetting software) isn’t so easy when you’re also trying to write a 5-page essay with a group and submit a data analysis for the lab,” they wrote.

Nevertheless, they had the motivation to finish what they started. “It was important for me to publish our evidence to confirm that our work is correct and worthy of respect,” says Johnson.

According to Jackson and Johnson, trigonometric terms can be defined in two different ways, and this can complicate attempts to prove the Pythagorean theorem. Focusing on just one of these methods, they developed four proofs for right triangles with sides of different lengths and one for right triangles with two equal sides.

Among them, one evidence stands out Lozano-Robledo. In it, students fill one larger triangle with an infinite sequence of smaller triangles and use calculus to find the side lengths of the larger triangle. “It’s like nothing I’ve ever seen,” says Lozano-Robledo.

Jackson and Johnson also leave five more pieces of evidence “for the interested reader to peruse,” they wrote. The paper contains a lemma — a kind of stepping stone to the proof of a theorem — that “provides a clear direction for additional proofs,” Johnson says.

Now that the evidence is published, “other people can take the paper and generalize that evidence, or generalize their ideas, or use their ideas in a different way,” says Lozano-Robledo. “It just opens up a lot of mathematical conversations.”

Jackson hopes the publication of the article will inspire other students to “see that obstacles are part of the process. Stick to it, and you may achieve more than you thought possible.”