No One in Mathematics Can Prove That 10 Is a Solitary Number

If you're not a math fanatic, you might think you could figure this problem out. Spoiler: You can't. Proving that 10 is a solitary number has shown to be an impossible feat ... so far. How is this so difficult?!

I Can Tell That We Are Gonna Be Friends

Before you can even attempt to prove 10 is a solitary number, you need to really understand what a solitary number even is. A solitary number is a number that doesn't have any friends. In other words, if a number has friends, it's a friendly number, and if it doesn't, it's a solitary number.

Now, a friendly number is a number that shares a relationship with another number (either in a friendly pair or friendly n-tuple). Cute, right? You and your friends might share a relationship over your common love of Keanu Reeves films, but friendly numbers share a relationship over a common abundancy index. To find a number's abundancy index, you add the sum of the number's factors (that is, the numbers it can be divided by) and divide that sum by the number you started with.

Let's use 6 as an example. Adding the factors of 6 will give you 1+2+3+6, or 12. Now divide that sum (12) by the original number (6), and — ta-da! — your abundancy index is 2. The numbers 6 and 28 are a friendly pair because they both have an abundancy index of 2. (Breaking it down for 28 looks like this: 1+2+4+7+14+28 = 56, 56/28 = 2.) Fun fact: All perfect numbers are friendly and have an abundancy of 2, and 6 is the smallest friendly number. Got it? Good.

Why So Solitary?

Not all numbers are friendly (dun, dun, DUNNN!). In fact, all primes and prime powers are solitary, meaning they have no friends. In other words, running that little divisor exercise on solitary numbers will give you an abundancy index that's unique.

This brings us back to the original matter at hand: 10. It seems like 10, the number upon which the entire metric system is based, would be an easy nut to crack. Weirdly, it hasn't been so easy to find another number that shares 10's abundancy index of 9/5. At the very least, we know that 10 has no friends that are less than 2,000,000,000 (that's two billion). So, that's a start?

But 10 is definitely not alone in this unsolvable group: It's believed that 14, 15, 20, 22, 26, 33, 34, 38, 44, 46, 51, 54, 58, 62, 68, 69, 70, 72, 74, 76, 82, 86, 87, 88, 90, 91, 92, 94, 95, 99, 104, 105, 106, and many others are also solitary, but proving it appears to be extremely difficult. Proving that 10 — or any of the others — is a solitary number is what's called an open problem, an unsolved problem in mathematics. To make matters even more complicated, there are only a handful of numbers known to be solitary: 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369 ...

Why is this so hard? Well, proving there are no friends of 10 is kind of like proving that there are no purple swans. It's easy to prove it wrong — the minute you find one, you're done — but not so easy to prove it right. The number of numbers out there is considerable. It's, you know, infinite.

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Want more unsolved math? Check out Keith J. Devlin's "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time." We handpick reading recommendations we think you may like. If you choose to make a purchase through that link, Curiosity will get a share of the sale.

Written by Joanie Faletto February 23, 2018

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