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Mathematics and Computer Science

Treviño and student get published

Assistant Professor of Mathematics Enrique Treviño and Ugur C. Cengiz ’17, together with Paul Pollack of the University of Georgia, had a paper accepted for publication in a specialized mathematics peer-reviewed journal.

Cengiz worked with Treviño on a Richter project concerning perfect polynomials in the summer of 2014. Together with Pollack, an Associate Professor of Mathematics, the trio submitted “Counting Perfect Polynomials” to the prestigious journal Finite Fields and Their Applications

A number is called perfect if it is the sum of its proper divisors. For example, the proper divisors of 6 are 1, 2, and 3. We know 6 = 1 + 2 + 3, so 6 is a perfect number. The next perfect number is 28 because 28 = 1 + 2 + 4 + 7 + 14. There are 49 known perfect numbers and all of them are even. It is conjectured that there are no odd perfect numbers, but mathematicians have been unable to prove it. We do know that the first 100…00 (with 300 zeroes) numbers do not contain an odd perfect number.

There is a way to generalize this concept of perfect numbers to perfect polynomials. It is also believed that there are no odd perfect polynomials. Cengiz, together with Treviño and Pollack, showed that the first 100…00 (with 60 zeroes) polynomials do not contain an odd perfect polynomial. The paper proves other technical results. Their research expanded on work by Canaday, a mathematician from the early 20th century.

In a 2014 summer report Cengiz wrote “working with Dr. Treviño on various proofs and understanding Canaday’s paper was priceless. I don’t think a college student gets a lot of opportunities to work one-on-one with a professor and ask any question. Being able to understand even only portions of Canaday’s paper made me develop a keener curiosity for mathematics. I definitely will be working on more math problems in the future.”

Cengiz will pursue a master’s in mathematics at Bilkent University in Turkey starting this fall.

A preprint for the paper can be found on Professor Treviño’s website.