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Why Number Theorists Tackle Centuries-Old Problems

Photos by Flickr, Graphic by Jessica Olsen

Dr. Roger Baker’s office whiteboard fills up in twenty minutes with centuries-old problems for which no mathematician has an answer or a proof.

“All of these, you can describe them to anybody who understands high school math, really,” Baker said. “That’s why I love number theory. You can describe the problems, but they are incredibly hard.”

Number theory is the study of integers. Number theorists examine prime numbers, fractions, and numerical patterns that often appear random.

Across his whiteboard, Baker has written the names of famous mathematicians—Gauss, Maynard, Goldbach, Mersenne—and the simple statements they made that have left number theorists scratching their heads for decades. How many prime numbers can be found in any given range? How many of these prime numbers are “twin primes” (primes that are consecutive odd numbers)?

“Nobody has known how to solve these yet, even though in some cases there’s a million-dollar prize,” Baker said. “Certainly fame and fortune.”

And then there is the biggest question.

“The oldest problem in number theory:  Are there any odd perfect numbers [a number whose factors add up to equal the original number]? It’s that easy to create—to write down—a problem that we can’t do,” Baker said.

While certain branches of number theory have applications in security and encryption, Baker’s work is purely “for aesthetics.” His work has no practical application—for now.

“One of the wonderful things about math is that we can’t predict when there are going to be practical applications,” Baker said. “Suddenly somebody will put something together, a bit of this, a bit of that. . . . If you ask me again in twenty years, it will be a different answer.”

One of the highlights of nearly half a century exploring number theory has been his work on fractional parts. A fractional part is the string of digits to the right of a number’s decimal point: think the “.14159” portion of pi’s “3.14159.”

“If you have a sequence of fractional parts, how does it behave?” Baker said. “Your gut feeling is the fractional parts are fairly random . . . and that they must hit every interval between 0 and 1, with a frequency corresponding to its length. It gets a bit technical if I try to explain it,” Baker added with a chuckle.

Baker is the senior scientist in CPMS, having published his first peer-reviewed article in 1970. The fun he has ruminating over the riddles of number theory has kept him motivated over the past forty-seven years.

“We do it for fun,” Baker said. “Not for promotions, not for prizes. We do it because it is fun.”

“It’s one of the nice things about science, generally,” Baker said. “Doing it even when you start to get a bit feeble and using a walking frame to get around, you’re still writing beautiful papers that other people couldn’t have written. That’s pretty cool.”