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The Rare Beauty of Perfect Numbers

by | Aug 17, 2021

Perfection exists when something cannot be improved — when it has no flaws or defects. Few things in life are considered perfect. Sometimes perfection is subjective, like Nadia Comăneci’s perfect score on the uneven bars at the 1976 Olympics or Pitchfork’s rating of Animal Collective’s song What Would I Want? Sky as a 10/10. Other types of perfection are objective, like bowling 300 or having perfect pitch like Mariah Carey and Jimi Hendrix.

“Perfect Numbers” are a different type of perfect – they are “perfect” tautologically, meaning that their perfection lies in fitting a particular mathematical definition.

A perfect number is a positive integer that is the sum of its positive divisors excluding itself. For example, 6 is a perfect number as its factors, 1, 2, and 3, sum to 6.

28 is also perfect as 1 + 2 + 4 + 7 + 14 = 28.

Here are a few interesting things about perfect numbers:

• There are only four perfect numbers below 33 million: 6, 28, 496, and 8,128. The fifth perfect number is 33,550,336.
• Only 51 perfect numbers have been discovered thus far, but the search is on for more. Number 51 was discovered in 2018 and has 49,724,095 digits.
• No odd perfect numbers have been discovered. Some have theorized that odd perfect numbers are impossible, but there has been no definitive mathematical proof of this. However, it has been proven that prime numbers can’t be perfect.

Most of the other interesting things about perfect numbers are beyond my mathematical understanding.

So, what use are perfect numbers? They aren’t of practical use. Rather, as one mathematician put it, “Perfect numbers create a ‘playground’ for the interested.” The concept of perfect numbers doesn’t interact with other areas of mathematics – they are just an interesting component of number theory.

Here’s a few IFODs about numbers that you may find interesting:

The Interesting Number Paradox (which was the first IFOD I published)

Good News – Prime Numbers Go All the Way Up – This is about Euclid’s proof that primes are infinite. One mathematician has commented that “not knowing of this proof is a gap of experience in your life equivalent to having never tasted chocolate or having heard a single piece of music.”

Was Math Invented or Discovered – I think this is one of the most interesting IFODs I’ve written.

The History of Numbers – numbers haven’t always existed. Our ancient ancestors couldn’t even count!

Benford’s Law – This is pretty interesting.

1. I always look forward to your posts. I feel like I always learn something new or learn more detail about something I (thought) I knew. Thank you!

• Thanks! That’s so nice.

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