Alter Ipse Amicus

It was during the first year of my college, that I learned about a day dedicated for friendship – the first Sunday of August – Friendship Day. Every year, in our college hostel, a big group of friends got together and planned games, outings and many other exciting activities for Friendship Day. Shopping for gifts and cards for our best buddies at the Archies card store was one of the highlights during this time, and this remains one of my fondest memories of college even today.

As our college life came to an end, we were uprooted and planted in different parts of the world. Priorities changed but the bond of friendship still remained intact. Thanks to technology and prolific use of social media, we are still in touch with each other at the touch of a button; we are ready-as-always for pranks and games.

Yet another August is here and social media is already flooded with quotes and wishes for Friendship Day. I was looking for an interesting game to share and play with my college-mates, when I stumbled upon a Latin quote – “Alter Ipse Amicus”. A quick web search led me to its meaning – “friend is another self”. The quote sounded philosophical to me, until I could associate a mathematical meaning to it.

Friendship and Mathematics! Is there a relation? I set out to find this relation and VOILA! – I came across “Amicable Numbers” or “Friendly Numbers”.

Two numbers are called amicable numbers, when the sum of the aliquot divisors (all divisors excluding the number itself) of one number equals to the other.

Let us look at an example:

220 – Aliquot divisors are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110

284 – Aliquot divisors are 1, 2, 4, 71, and 142

The sum of Aliquot divisors of 220 = 1+2+4+5+10+11+20+ 22+ 44+ 55+110 = 284

The sum of Aliquot divisors of 284 = 1+2+4+71+142 = 220

220 and 284 are the smallest and earliest known pair of amicable numbers. This pair was originally found by Pythagoras. In 1636, Fermat discovered the pair 17,296 and 18,416, from ancient Arab literature. In 1638, Descartes found 93,63,584, 94,37,056. By 1747, Euler found around 30 pairs of amicable numbers. In 1866, a 16-year old school boy B. Paganini found the second smallest amicable pair 1184, 1210.

The amicable pair, 220 and 284 were regarded as a symbol of friendship in the Pythagorean brotherhood.  During the middle ages, this pair was used in making and matching of horoscopes. Talismans inscribed with 220 and 284 were believed to promote love.

In the 20th and 21st centuries, Mathematicians have generated numerous amicable pairs, using computer programmes. Yet, it is unknown whether infinitely many pairs of amicable numbers exist.

I think in the context of amicable numbers, the quote “Alter Ipse Amicus” makes perfect sense. Mathematicians’ love of patterns in numbers is long lasting and so is the strong bond of friendship among friends.

Happy Friendship Day to All !

– Jyothi Giri, Facilitator, BrainSTARS

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