The Undiscovered Perfect Number
Tina Xia
I was poking around on WolframAlpha the other day and came across an article called, “Perfect Numbers”. Between the page’s strange jargon, I managed to pick apart the concept, and reach a solid, albeit surface-level, understanding of what a perfect number is, and thought it was cool enough to share. I hope when you read through this, you’re at least a little bit unnerved by how beautifully math can work out in its infinite patterns.
Perfect Numbers
So, to begin, consider the number 6. If you list out it’s factors — 1, 2, 3, and 6 — then sum the numbers excluding 6 itself, you’ll end up with 6 again. And there we have the smallest perfect number—positive integers whose factors, excluding the original integer, sum up to itself. After 6, the next perfect number is 28, whose proper factors 1, 2, 4, 7, and 14, sum 28.
Laying out the first handful of perfect numbers, you get something like this:
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
Mersenne Primes
To fully understand perfect numbers, a group of numbers called Mersenne primes need to be considered as well. They can be categorized as prime numbers in the form 2n-1 for any number n. As you can probably predict, not every number plugged into n will return a prime number, but through trial and error, the first few can be found.
Taking 1 to be n, for example, returns 1, which isn’t a prime number, but taking 2 to be n, returns 3, which is a prime number. To make the pattern clear, the first few natural numbers taken as values for n and the numbers they return are laid out in the following chart. Prime outputs are bolded.
If we take these Mersenne primes and compare them to our list of perfect numbers, we can pick out a connection. The same list from before is copied down below, but this time, with some notable values bolded.
6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064
Evidently, a Mersenne prime exists in the factors of every perfect number. In fact, they appear in corresponding order with each other. The first perfect number contains the first Mersenne prime in it’s factors, the second perfect number contains the second Mersenne prime, and so on and so forth. Really, Mersenne primes are the reasons we ever discovered perfect numbers, and our guides to tracking down new ones. Back in 2018, a perfect number 49,724,095 digits long was found this way.
The Odd Perfect Number
Okay, one more thing to leave you with. The whole Mersenne prime connection only applies to even perfect numbers. This doesn’t really matter since every perfect number we know of is even—no one has ever discovered an odd perfect number. The thing is, though, no one has ever proven that odd perfect numbers can’t exist. There could very well exist an odd perfect number, that we just haven’t discovered quite yet. But, we do know this: if we ever come across one, it’s factors won’t include a Mersenne prime.
Works Cited
“Odd Perfect Number -- from Wolfram MathWorld.” Wolfram MathWorld, https://mathworld.wolfram.com/OddPerfectNumber.html. Accessed 16 February 2022.
“Perfect Number -- from Wolfram MathWorld.” Wolfram MathWorld, https://mathworld.wolfram.com/PerfectNumber.html. Accessed 16 February 2022.
Perfect Numbers and Mersenne Primes - Numberphile. Performance by Matt Parker, Numberphile, 2015. YouTube, https://www.youtube.com/watch?v=T0xKHwQH-4I. Accessed 16 February 2022.
Page layout by Tina Xia