120. Friendly Numbers


Friendly Numbers are amicable numbers! According to Pythagoras, the Greek Mathematician, two numbers are friendly if each of them is equal to the sum of the divisors of the other number. An example will make the meaning clearer.

The Greeks knew the first pair of Friendly Numbers and they are 220 and 284.
The divisors of 220 namely 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 add up to 284!
The divisors of 284 namely 1, 2, 4, 71 and 142 add up to 220!

In 1636, another pair of Friendly Numbers was discovered by the French Mathematician Pierre de Fermat. The numbers were 17296 and 18416.

By the middle of the 19th century, more than 60 pairs of Friendly Numbers had been discovered. Surprisingly the second lowest pair of Friendly Numbers had been over looked by everyone till 1867.

A sixteen year old Italian, Nicolo Paganini, proved that 1184 and 1210 are the second lowest pair of Friendly Numbers, in 1867.

With the help of computers, the list of Friendly Numbers keeps growing all the time. But still many doubts haunt the minds of the mathematicians.

Why the two numbers in any pair are either odd numbers or even numbers?
Can we have a pair with one odd number and one even number?
Why all the odd numbers are multiples of 3?

These queries continue to puzzle and challenge the minds of the mathematicians all over the world!

Visalakshi Ramani

41 thoughts on “120. Friendly Numbers”

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