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.
A perfect number is a number where the sum of all the divisors, excluding the number, are equal to the number itself
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!
Sound Bites
• Mathematician Michel Talagrand from the French National Centre for Scientific Research (CNRS) has won the 2024 Abel prize. This prize is sometimes called the Nobel prize of mathematics. Talagrand received the prize for his contributions to probability theory, stochastic systems and functional analysis, with important applications in mathematical physics and statistics. The Norwegian Academy of Science and Letters in Oslo announced the winner of the prize prize on March 20.
• In 2021, Giorgio Parisi shared the Nobel Prize in Physics for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales. Talagrand and others provided the mathematical grounds for spin glasses that were pivotal in Parisi’s work.
DID YOU KNOW?
• The sum of the reciprocals of the divisors of a perfect number add up to two. For example, 1/1 + 1/2 + 1/3 + 1/4 + 1/7 + 1/14 + 1/28 = 2.
• The binary representation of a perfect number is formed by n ones followed by n-1 zeros for some n. For example, 111110000 is the binary representation of 496.
• Amicable numbers are pairs of numbers in which the sum of the proper divisors of one is the other. For example, 220 and 280 is a pair of amicable numbers. It is not known whether there is an infinite number of amicable pairs or if there is a pair formed by one odd and one even number.
• A sequence of n numbers is called sociable if the sum of the proper divisors of the first number is the second number, the proper divisors of the second add to the third, and so on until it loops to the first. For example, 12,496, 14,288, 15,472, 14,536, 14,264 is a sequence of five sociable numbers. To date, no sequence of sociable numbers of size 3 have been found.