The perfect problem
Mathematicians do their best to keep competition to a minimum by requiring years of study before they finally reach the point where there is something new and interesting to be discovered. Fortunately, there are still a few very interesting problems that can be solved by a clever soul with just a pencil, a sheet of paper and some time on their hands.
A perfect number is a number where the sum of all the divisors, excluding the number, are equal to the number itself. For example, six is a perfect number because the divisors one, two and three sum to six. The first two perfect numbers are six and 28.
If you’re not quite sure it’s a coincidence that God created the world in six days and the moon orbits the Earth every 28 days, then you’d be in good company. St Augustine had similar thoughts.
Before Christ walked the Earth, people have been searching for these perfect numbers and had found two additional ones: 496 and 8,128. These numbers were known by the Greek mathematicians Pythagoras and Euclid.
The discovery of other perfect numbers stalled until the 13th century when Egyptian mathematician Ismail ibn Fallūs identified the next three perfect numbers: 33,550,336, 8,589,869,056 and 137,438,691,328.
With the dawn of computers, it became possible to find many more that would be entirely impractical to calculate by hand. Mathematicians can’t compete with computers on computation, so instead they always focus on problems that computers can’t solve. A bit like when someone challenges me to a running race, I turn to an obscure math problem.
Oddly, all discovered perfect numbers are even. Mathematicians aren’t sure why, and they also aren’t sure if there are infinitely many perfect numbers. This, dear reader, is where you step in to save the day!
1. Are there any odd perfect numbers?
2. Are there infinitely many perfect numbers?
Anyone with high school mathematics has the tools to solve these problems. In mathematics, the challenge is almost never the actual computation. It is the idea that sparks the understanding that leads to the solution.
Henry Ford wanted curved windshields for his vehicles. His glass workers told him this was impossible, so he challenged several young engineers with no glass working experience to create a curved windshield. They had the solution in under a week.
I would encourage you to work on this problem without looking up other people’s attempts at solving it.
Perhaps by starting with a blank slate, you might beat the mathematicians at their own game ‒ and in the process become one!