Summer Research Yields Prime Results

Jul 20, 2018


Adrian Rice and Martha HarttThanks to R-MC's Schapiro Undergraduate Research Fellowship (SURF) program, Martha Hartt '20, a mathematics and computer science major, is researching a subject she is passionate about: The Distribution of Primes. SURF gives students the opportunity to conduct 10 weeks of research under the guidance of a faculty member. Hartt is working under the mentorship of Mathematics Professor Adrian Rice.

Prime Research
Prime numbers (positive whole numbers only divisible by themselves and 1) are scattered completely randomly throughout the number system.

"Initially they are quite easy to find (e.g., 2, 3, 5, 7, 11, ...), but as you get higher, they get far more erratically spaced apart," explains Rice. "It has been known for over 2,000 years that there are infinitely many prime numbers, but one of the hardest problems in math at the moment is to find some sort of rule or pattern by which higher prime numbers can be located. The project on which Martha is working is one of the many smaller problems which will help to narrow down our knowledge of how prime numbers are distributed."

Creative Solutions
"Mathematics research generally entails attempting to definitively prove 'things,'" says Hartt. "These 'things,' when proven, become theorems. Proof requires a lot more than simply showing that the theorem works for a lot of values. To prove a theorem in math, it's not good enough to show that it works hundreds, even thousands, of times: You need to show that it is always true—period. I have been using a variety of general theoretical techniques derived from several of my math classes to attempt to prove my final result."

Through her research, Hartt hopes to expand her knowledge of proof strategies and improve her ability to assess a problem and use an appropriate strategy to attempt to find a solution.

"My project consists of many small lemmas (mini-theorems), which, when combined, will produce the main result," she explains. On a typical day of research, Hartt examines a lemma and uses different strategies to attempt to prove it. She also does what she calls "looking at big-picture stuff"—assessing what information is actually necessary, and which ideas have hit dead ends.

Unique Approaches
Conducting research is a challenging and rewarding experience that highlights the importance of critical-thinking and problem-solving skills.

"It's a great experience because it allows me to do independent research, and it puts me in a situation where the 'correct' answer is not always clear," says Hartt, a Presidential Scholar. "The SURF program differs from my math classes in that I am not taught how to obtain my desired result; I am required to come up with an approach on my own. I also have plenty of opportunities to try different strategies, which further enhances my problem-solving skills."

The Big Picture
Hartt and Rice meet every other day to discuss the progress she has made.

Hartt, president of the R-MC chapter of Pi Mu Epsilon, the national mathematics honorary society, says, "Professor Rice gives me suggestions about which direction to take if I am stuck. This is very helpful, because often there are formulas or theorems out there that could help me with my research that I don't know about and that I wouldn't even think to consider using. Professor Rice also helps me see the big picture."

Rice says, "Martha is a pleasure to supervise because she is a terrific student who is passionate about mathematics. She has the scholarly independence and tenacity required to attack a problem until she has either solved it or worked out that a different approach is needed. She has already had some very good ideas and will undoubtedly contribute several more before we're done. I think we're going to end up with a very fine piece of work by the time we're finished."