Two teenagers have once again proven an age-old math rule

Two years ago, a few high school classmates each wrote a mathematical marvel, a trigonometric proof of the Pythagorean theorem. Now they’re unveiling ten more.

For more than 2,000 years, such proofs were considered impossible. And yet Ne’Kiya Jackson and Calcea Johnson published, undeterred their new evidence October 28th American Mathematical Monthly.

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

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

Mathematical proofs are sets of statements that prove a statement is true or false. The Pythagorean Theorem – a² + b² = c²related the length from the hypotenuse of a right triangle to that of the other two sides — has been proven many times with algebra and geometry (SN: 2/4/03).

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

Jackson and Johnson came up with the first of their trigonometry-based proofs in 2022, while they were seniors at St. Mary’s Academy in New Orleans, a Catholic school attended primarily by young black women. At the time, only two other trigonometric proofs of the Pythagorean theorem existed, presented by mathematicians Jason Zimba and Nuno Luzia in 2009 and 2015, respectively. Working on the first proofs “sparked the creative process,” says Jackson, ” and from there we developed additional proofs.”

After formal present their work At a meeting of the American Mathematical Society in March 2023, the duo wanted to publish their findings in a peer-reviewed journal. “This turned out to be the most difficult task of all,” they said in the newspaper. In addition to writing, the duo had to develop new skills while attending college. “Learning to code in LaTeX (a typesetting software) is not that easy when you are also trying to write a five-page essay with a group and submit a data analysis for a lab,” they wrote.

Still, they were motivated to finish what they started. “It was important to me that our proofs be published to confirm that our work is correct and respectable,” 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.

One proof of this stands out to Lozano-Robledo. In it, students fill a larger triangle with an infinite series of smaller triangles and use calculus to find the side lengths of the larger triangle. “It looks like nothing I’ve ever seen,” Lozano-Robledo says.

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

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

Jackson hopes the publication of the article will inspire other students to “realize that obstacles are part of the process. If you persevere, you will find that you achieve more than you thought possible.”