Friends and Mathematician Ramanujan

Mathematician Srinawasan Ramanujan didn’t have any close friends- someone asked him the reason. He replied that although he wanted to have close friends -nobody was up to his expectations. When pressed how he expected his friend to be – he replied – like numbers 220 and 284!

The person got confused and asked what is the connection between friendship and these numbers!

 

Ramanujan asked him to find the divisors of each number!
With much difficulty – the person derived and listed them

220 → 1,2,4,5,10,11,20,22,44,55,110,220

284 →1,2,4,71,142,284

Ramanujan then asked the person to exclude the numbers 220 and 284 and asked the sum of the remaining divisors

The person was astonished to find:

220 → 1+2+4+5+10+11+20+22+44+55+110=284

284 →1+2+4+71+142=220

Ramanujan  explained that an ideal friendship should be like these numbers- to complement each other – even when one is absent – the other should represent the friend!

The person thought – no wonder this genius is on the world’s top list of mathematicians!!

 

Hardy–Ramanujan number 1729

The number 1729 is known as the Hardy–Ramanujan number after a famous visit by Hardy to see Ramanujan at a hospital. In Hardy’s words:

I remember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. “No”, he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

Immediately before this anecdote, Hardy quoted Littlewood as saying, “Every positive integer was one of [Ramanujan’s] personal friends.”

The two different ways are

1729 = 13 + 123 = 93 + 103

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