Students discover and publish unexpected proof for 2,000-year-old mathematical theory

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Louisiana students Ne’Kiya Jackson and Calcea Johnson wowed their teachers in 2022 when they discovered a new way to prove the 2000-year-old Pythagorean theorem in response to a bonus question in a high school math contest. But that was only the beginning.

A volunteer at their former school, New Orleans’ St. Mary’s Academy, encouraged them to submit their work on the famous mathematical theory to a professional conference, and in March 2023 they became the youngest people to present at the American Mathematical Society’s Southeastern Sectional conference in Atlanta. Their appearance elicited a wave of media coverage, including a spot on “60 Minutes.” The pair also received symbolic keys to the city of New Orleans and a shout-out from Michelle Obama.

Now Jackson and Johnson, who started college last year, have notched another achievement: authoring an academic paper detailing their original proof — plus nine more. Their work published Monday in the scientific journal American Mathematical Monthly.

“I’m so surprised that we’re getting published in a paper at such a young age,” Johnson, a sophomore studying environmental engineering at Louisiana State University, said in a video shared by the journal publishers.

“I didn’t think it would go this far,” said Jackson, who is pursuing a doctoral degree in pharmacy at Xavier University of Louisiana.

Pythagoras was an ancient Greek philosopher and mathematician who lived 2,500 years ago. It’s not clear whether he or his disciples came up with his eponymous theorem. But the principle is a mainstay of math class and enables the calculation of the length of any side of a right-angled triangle — as long as the lengths of the other two sides are known. It’s usually summarized as a^2 + b^2 = c^2, in which the square of the longest side (the hypotenuse) is equal to the squares of the two shorter sides added together.

Many mathematicians over the years have explained why the theorem works using algebra and geometry. In contrast, Jackson and Johnson were able to prove it using trigonometry — a subbranch of mathematics that focuses on the study of triangles.

Doing the ‘impossible’

Experts described Jackson and Johnson’s approach as particularly challenging because trigonometry as a field is essentially based on Pythagoras’ theorem; thus using trigonometry to prove the theorem typically requires what mathematicians call “circular reasoning.” However, the new proofs aren’t circular, according to the study.

“None of the theorems we use in our proofs … have already assumed the Pythagorean theorem to be true,” the authors wrote in the paper.

Tom Murdoch, an honorary professor at the University of Bristol’s School of Mathematics in the United Kingdom, described the study as “impressive.”

“I think what’s intriguing about it (this study) is that a lot of people thought it was impossible,” he said.

“Trigonometric functions are based on sine and cosine, which are expressed as ratios of certain lengths of a right-angled triangle. It’s quite easy to get into a circular argument, and what’s so appealing about this is they found a line of argument using sine and cosine that doesn’t assume Pythagoras is true,” he added.

Jackson and Johnson’s study outlines five new ways of proving the theorem using trigonometry, and their method reveals five more proofs for 10 total. The pair presented only one of the proofs at the 2023 conference, meaning that nine are totally new.

“I think sometimes having very little knowledge of the problem means that you are not bound by what’s gone before. Looking at this with fresh eyes, which I think they’ve done, that’s the really impressive thing,” Murdoch said.

Preparing the academic paper while they started college was “the most daunting task of all, since we had absolutely no experience writing for an academic journal,” the pair noted in the study. Their road to publication included a standard hurdle known as peer review, or the stage during which eminent scientists in the field examine and comment on the work.

Johnson said she was happy to be able to show women and women of color in particular what can be achieved in a traditionally male-dominated field.

“I am very proud that we are both able to be such a positive influence in showing that young women and women of color can do these things, and to let other young women know that they are able to do whatever they want to do. So that makes me very proud to be able to be in that position,” Johnson said in a news release.

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