A glorious marriage

The number 1148’s family has 11 members: {1, 2, 4, 7, 14, 28, 41, 82, 164, 287 and 574}.  

Each of them, like a child running into the warm enfolds of his mother’s arms, is a perfect divisor of the number 1148: In every instance, when 1148 is divided by any of these numbers, the result is a whole number. There are no limbs sticking out of this hug, and the hug is wide enough to be able to accommodate each number no less than twice and up to 1148 times.  

The number 1210’s family also has 11 members: {1, 2, 5, 10, 11, 22, 55, 110, 121, 242 and 605}.  

Just like in the number 1148’s family, each of these numbers is a perfect divisor of the number 1210. In this family, the arms of the number 1210 are wide enough to accommodate any of these numbers no less than twice and up to 1210 times.  

It has been known for many hundreds of years now that the numbers 1148 are 1210 are lovers. With 62 numbers between them on the number-line, it was realized that these numbers had travelled many miles, across great seas and oceans and over lofty mountain ranges, just to be with each other. Eventually, by sheer coincidence, the 2 numbers found a way. It was found by a famous Indian mathematician that if you found the sum of the factors of 1210, the result should be, if mathematics does not bully you the way it bullies me, 1148. The opposite is equally true: The sum of the factors of 1148 is 1210. And so, in a glorious marriage, the two found a way to be together:

  

1+2+5+10+11+22+55+110+121+242+605 = 1148 

1+2+4+7+14+28+41+82+164+287+574 = 1210 

The word that was chosen to describe these two numbers is a monumental understatement of what it means for such a relationship to be discovered. The pair of numbers 1148 and 1210 is said to be an amicable pair. The smallest such pair is the numbers 220 and 284. Amicable pairs make up one half of a small family of numbers that are described as sociable.  

The other half of this family is made up of perfect numbers. These numbers form very tight knit families. The sum of the children of a perfect number is the perfect number itself. 28, for example, has 5 children: 1, 2, 4, 7 and 14. 1+2+4+7+1 = 28. The number 28 is also said to be perfect because it can be expressed as the sum of consecutive integers: 1+2+3+4+5+6+7 = 28...

 

Since time immemorial, our species has made an active habit of looking up at the stars and out into the universe beyond. Homo erectus, our first ancestor to stand upright, likely stared in wonder at the heavens, wondering how the sky had managed to ignite tiny, eternal flames when he had failed to ignite a campfire that would never die. Homo sapiens neanderthelensis likely looked with gratitude at the moon that illuminated the land at night, revealing dangerous animals from the nooks and crannies in which they hid. 

Homo sapiens, however, were the first to look at the stars and see art, in the form of constellations. We looked up and insisted we had seen a plough in the stars. We looked up and claimed there was a lion staring down at us from up above. But most importantly, we looked up at the sky and saw all of our heroes, from Orion, to Hercules, and many, many others. We looked up at the sky and saw ourselves in the stars.  

We looked up at the sky and saw ourselves, because we could not see ourselves elsewhere.  

And loneliness could never have been the birth-right of a species whose defining characteristic is community.   

And so, we looked, and looked, and looked, and found ourselves everywhere. We heard the haunting sound of the human voice in musical instruments that we insisted were ‘singing’. We saw barter trade, the most basic system for the distribution of goods, in the atoms that we described as having ‘swapped’ electrons.  

We fashioned our existences, the stories we told and the language we used, to reflect us, our history, and the things that we value the most... 

The sociable numbers, aside from being incredibly rare (no definite proof exists to show that an infinite number of them exists), are, together with constellations, a beautiful example of what happens when a complex species such as our own finds itself exposed to the mysteries of the universe and instructed to understand them for its own sake, at its own risk. They are a reminder of the poetic loneliness behind how our species has chosen to understand the universe: To understand pairs of cold, unflinching numbers as loving companions and to understand arbitrarily scattered stars as portraits of the heroes of old.  

And yet, they are a tragic reminder of how willing we are to understand the world in human terms, except where it matters the most: With other humans. Humans who may not speak the language we speak, or call God by the names we use. Humans with different coloured skin and humans from different lands. They are humans whom we have chosen to see, not as companions to be loved, the way we have seen amicable pairs love, but as enemies to be destroyed.  

They are humans we have chosen to see as evil... 

An abundant number is a number for which the sum of its factors is greater than the number itself. It is not perfect, and can never be perfect. A deficient number is a number for which the sum of its factors is less than the number itself. It, also, is not perfect and can never be perfect.  

My favourite pair of these two sets of numbers is 20 and 14: 

The factors of 20 are 1, 2, 4, 5 and 10. The sum of these factors is 22. The number 20 is abundant.  

The factors of 14 are 1, 2 and 7. The sum of these factors is 10. The number 14 is deficient.  

We are all imperfect numbers, with a little too much of this and not enough of that, thrown together by the randomness of existence in little amalgamations called societies.  

Yes. That’s where this is going.  

But we are imperfect numbers thrown together in the hope of creating perfection. The perfection of sociability, amongst ourselves and with people who may see life a little different from us. The perfection of hope, for those who may have never seen life for what it truly is. The perfection of love.  

Perfection such as in the number 6, the smallest perfect number, whose factors (1, 2 and 3) all add up to 6.  

 

20 – 14 = 6 

Sincerely,  

Muku 

Photo by Annie Spratt on Unsplash

Inspired by: The Housekeeper and the Professor, by Yoko Ogawa