The Quest to Find Odd Perfect Numbers

I'm not sure why I find articles about advanced math and number theory so interesting. I was something of a math whiz when I was a teenager and took two years of math and physics in university before running into tensor calculus and switching to English literature. But I still enjoy reading about some of the more arcane aspects of math, and number theory certainly fits that bill. 

So here's an article about mathematicians quest to find odd perfect numbers or alternatively to prove that there aren't any. 

AS A HIGH school student in the mid-1990s, Pace Nielsen encountered a mathematical question that he’s still struggling with to this day. But he doesn’t feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

Part of this problem’s long-standing allure stems from the simplicity of the underlying concept: A number is perfect if it is a positive integer, n, whose divisors add up to exactly twice the number itself, 2n. The first and simplest example is 6, since its divisors—1, 2, 3, and 6—add up to 12, or 2 times 6. Then comes 28, whose divisors of 1, 2, 4, 7, 14, and 28 add up to 56. The next examples are 496 and 8,128.

Leonhard Euler formalized this definition in the 1700s with the introduction of his sigma (σ) function, which sums the divisors of a number. Thus, for perfect numbers, σ(n) = 2n.

But Pythagoras was aware of perfect numbers back in 500 BCE, and two centuries later Euclid devised a formula for generating even perfect numbers. He showed that if p and 2p − 1 are prime numbers (whose only divisors are 1 and themselves), then 2p−1 × (2p − 1) is a perfect number. For example, if p is 2, the formula gives you 21 × (22 − 1) or 6, and if p is 3, you get 22 × (23 − 1) or 28 — the first two perfect numbers. Euler proved 2,000 years later that this formula actually generates every even perfect number, though it is still unknown whether the set of even perfect numbers is finite or infinite.

Nielsen, now a professor at Brigham Young University (BYU), was ensnared by a related question: Do any odd perfect numbers (OPNs) exist? The Greek mathematician Nicomachus declared around 100 CE that all perfect numbers must be even, but no one has ever proved that claim.

There probably is some practical application for this kind of number theory research, but that doesn't matter. It's just a real cool and deep intellectual exercise, and I find it fascinating that mathematicians have been working on it for millennia and still haven't come to a resolution.