Mathematicians Open a New Front on an Ancient Number Problem | Number Theory - Quanta Magazine

Steve Nadis, Contributing Writer observes, For millennia, mathematicians have wondered whether odd perfect numbers exist, establishing an extraordinary list of restrictions for the hypothetical objects in the process. Insight on this question could come from studying the next best things. 

A comically long “missing persons” poster for an odd perfect number that shows all the restrictions it has to satisfy.
Photo: Jason Chuang for Quanta Magazine

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..

Tantalizing Near Misses 
The first spoof was found in 1638 by René Descartes — among the first prominent mathematicians to consider that OPNs might actually exist. “I believe that Descartes was trying to find an odd perfect number, and his calculations led him to the first spoof number,” said William Banks, a number theorist at the University of Missouri. Descartes apparently held out hope that the number he crafted could be modified to produce a genuine OPN.

But before we dive into Descartes’ spoof, it’s helpful to learn a little more about how mathematicians describe perfect numbers.
Read more...

Source: Quanta Magazine