# Bucknell World: The beauty of numbers

(Editor's Note: Bucknell's Web site is featuring some of the University's newest teacher-scholars. They are among the new faculty members highlighted in the Fall 2007 edition of Bucknell World.)

The next time you make a secure online purchase, thank number theory. This little-understood branch of mathematics guides the cryptography that keeps your credit card information secret.

In talking to assistant professors of mathematics Sharon Garthwaite and Nathan Ryan, one senses they are grateful for this application, because it helps them to justify their field to non-mathematicians. However, for them, the pure intellectual beauty of the theory, and not potential applications, drives their research.

Modular forms
Both professors work with modular forms, which Ryan explains not by describing what they are, but how they ave been used. A famous mathematical puzzle, Fermat’s Last Conjecture, had stumped mathematicians since the 1600s: Why does the equation xn + yn = zn have no non-zero integer solutions when n is greater than 2?

“There are infinitely many solutions when you look at x2 + y2 = z2, but if you raise the exponent by one, there are no solutions,” Ryan said. “I think that is pretty amazing.”

In the 1990s, a Princeton mathematician used modular forms to prove why the equation has no solutions. The proof was announced during Ryan’s college days. He’s been hooked on mathematics ever since. Today, he uses his skills as a self-taught computer programmer to examine the computational aspects of modular forms.

Partition theory
Garthwaite studies partition theory. Explained simply, this theory encompasses the questions that arise from the numerous ways of partitioning whole numbers into smaller whole numbers.

Her research stems largely from the writings of Srinivasa Ramanujan, a self-taught Indian mathematician who left behind a wealth of tantalizing results when he died in 1920 at the age of 32.

“Close to 100 years later, we are still trying to figure out exactly what he knew and piece it together like a puzzle,” Garthwaite says.

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Contact: Office of Communications

Posted Nov. 28, 2007

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