The Y Report

Perfectly Odd

Portrait of Pace Nielsen.
Pace Nielsen. Photo by Bradley Slade.

In high school Pace P. Nielsen (BS ’01) stumbled upon a centuries-old unsolved math problem: the odd perfect number (OPN) conjecture. This problem, contemplated by Pythagoras and Euclid, helped spark Nielsen’s passion for theoretical math. Now a BYU math professor, Nielsen researches abstract algebra and, with the help of undergrads, has patiently chipped away at the OPN problem year after year—and has made some groundbreaking progress.

Perfect numbers have divisors that add up to twice their value. The divisors of 6, for example—1, 2, 3, and 6—add up to 12. All known perfect numbers are even—but are there any odd perfect numbers?

“It’s possible that one accidentally exists somewhere,” says Nielsen, who calls OPNs a “particularly beautiful problem.”

Mathematicians have unearthed conditions an OPN would have to meet. Some have even discovered rare “spoofs”—numbers that seem to fulfill these conditions but ultimately do not.

Nielsen and a BYU computational number theory team set out to find one spoof, maybe two. But “we found infinitely many,” says Nielsen. It’s a surprising discovery that forever changes how mathematicians approach OPNs.

Nielsen and his students haven’t given up on finding a true OPN. “You’re going to have failures,” he says. “You’re going to think about a problem for months or years. You’re going to have a breakthrough. And then you’ll start a new problem.”

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