In 2022, Louisiana high school students Ne’Kiya Jackson and Calcea Johnson amazed their teachers by finding a new method to prove the ancient Pythagorean theorem as part of a bonus question in a math competition. This was just the start of their journey.

A volunteer at their previous school, St. Mary’s Academy in New Orleans, encouraged them to present their findings on this renowned mathematical theorem at a professional conference. In March 2023, they became the youngest individuals to present at the American Mathematical Society’s Southeastern Sectional conference in Atlanta. Their participation garnered significant media attention, including a feature on 60 Minutes. Additionally, they received symbolic keys to the city of New Orleans and recognition from Michelle Obama.

Now in college, Jackson and Johnson have reached another milestone: publishing an academic paper that includes their original proof along with nine others. Their work appeared recently 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, mentioned 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 from 2,500 years ago. It remains uncertain whether he or his followers developed his well-known theorem. The principle is a staple in math classes, enabling the calculation of any side of a right-angled triangle if the lengths of the other two sides are known. It is often summarized as a^2 + b^2 = c^2, where the square of the hypotenuse equals the sum of the squares of the other two sides.

Over the years, numerous mathematicians have used algebra and geometry to explain why the theorem holds true. However, Jackson and Johnson proved it using trigonometry, a branch of mathematics that studies triangles.

Experts described their approach as particularly difficult since trigonometry is essentially rooted in Pythagoras’ theorem. Thus, using trigonometry to prove the theorem often involves what mathematicians call circular reasoning. According to the study, the new proofs avoid this circular reasoning.

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

Tom Murdoch, an honorary professor at the University of Bristol’s School of Mathematics, praised the study as impressive. “What’s intriguing about it is that many people thought it was impossible,” he said.

Jackson and Johnson’s study introduces five new methods for proving the theorem using trigonometry. Their approach uncovers five additional proofs, totaling ten. They presented only one of these proofs at the 2023 conference, meaning nine are entirely new.

“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 noted.

Johnson expressed her happiness in showcasing the achievements possible for women and women of color 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.