Randall Taylor Jones, who goes by his middle name, Taylor, is a graduate student in the UNT Mathematics department. Taylor's primary interest is in fractal geometry, but this past summer, he finished a research project with his professors Drs. Pieter Alaart, Stephen Jackson, and fellow graduate student David Lambert that veered from his usual studies. This number theory research focused on Trott Numbers, which are numbers whose continued fraction expansion agree with its decimal expansion.

"This was originally just a silly problem I thought of sometime in summer 2020, but when I could not solve it myself after a few hours of thinking, I searched about it online. While this idea has been thought of before, what was surprising is that no one has done any research what-so-ever about this topic," said Taylor. "Often in math, if a problem is easy to understand, then it is uninteresting, while if it is interesting, then it is not easy to understand. Therefore, if you find a problem that is both interesting and easy to understand, and that no one else has done before, then it is possible you have found a great opportunity." Taylor took the opportunity to work with a team of researchers, including undergraduate students he was mentoring, to tackle the problem.

Trott numbers have deep connections with Diophantine approximation theory, which is the study of how well (or unwell) real numbers can be approximated by rational numbers. "It's possible further advances in our problem could lead to advances on other problems in this field. I am hopeful that other math researchers will also want to research this topic, as there are still many interesting open problems left to solve," said Taylor.

Taylor recently gave a seminar talk regarding this research at the  One World Numeration Seminar, an online seminar on  numeration systems  and related topics. "After over a year of work it is nice be able to share our results, it's also very rewarding when mathematicians whose papers I have read and studied before are interested in something I worked on," he said.

Working as both a teacher and a student, Taylor says that finding the time and energy to prepare for classes and get his work done can sometimes be a challenge. Additionally, the Trott research project involved six people who needed to meet every week, but all of it was made possible with the support of his advisor, Dr. Pieter Allaart, and mentors, Dr. Kiko Kawamara and Dr. Lior Fishmen.

"Not only do each of these professors help me in math, but more importantly help me be a successful person at UNT," he said. "I am very lucky to have met them in life."

Since the project involved mentoring undergraduate students himself, it was a fun learning experience for Taylor, who also received his B.A. from the UNT Mathematics Department.

"I am surprised at how smart some students are, even though they only just finished high school!" he said. "The UNT math department is very good at encouraging bright undergrads to pursue math research. The barrier to entry into math research specifically can be very high, so it's a positive thing when undergrads who are interested in it can gain some early experience."

Learning to think independently about math and develop skills as a researcher and teacher have been invaluable experiences for Taylor as a graduate student. He wants to encourage younger students who are interested in mathematics research to follow their dreams and jump into it. "Talk to your professors about it!" he said. "The reason they are where they are is because they really enjoy math. If you are truly interested, they will be happy to lead you in the right direction. Also, study the basics to have a strong foundation. There is a good reason that musicians will practice scales and chords, or football players will lift weights; math is no different."

To learn more about the UNT Mathematics department and programs, visit math.unt.edu