ROMANCING WITH AMICABLE NUMBERS
V Phaneendrudu
Amicable numbers are two different numbers so related that
the sum of the proper divisors of each is equal to the other number. A proper
divisor of a number is a positive integer divisor other than the number itself.
For example the proper divisors of number 6 are 1, 2 and 3. The smallest pair
of amicable numbers is 220 and 284. The proper divisors of 220 are 1, 2, 4, 5,
10, 11, 20, 22, 44, 55 and 110. When all these are added, one gets 284.
Likewise when one adds the proper divisors of 284, viz. 1, 2, 4, 71 and 142 it
is 220.
Iamblichus of Chalsis (about A.D. 320) ascribes the knowledge
of amicable numbers to the earliest Pythogorean School (about 500 B.C.). The
amicable numbers played a role in magic and astrology, in the casting of
horoscopes, in sorcery, in the concoction of love potion and in the making of
‘talisman’. Persons who occupied them with talisman assured that these numbers
had a particular influence in establishing union and friendship between two
individuals. Through Arabs the knowledge of amicable numbers spread to Western
Europe. In the 9th Century Thabit Ibn Quarra (826 - 901) an Arab
mathematician invented first algebraic rule for amicable numbers. Thabit’s
formula was rediscovered by Fermat (1601-1665) and Descartes (1596-1650) and
extended by Euler (1707-1783). Prior to Euler only three amicable numbers were
only known. Euler was the first to study systematically the amicable numbers. He
developed several methods for generating amicable numbers. 59 new pairs of
amicable numbers were generated in his life time. A 16 year old Italian school
boy Nicolo Paganini found the second smallest pair of amicable numbers in 1866
(1184, 1210) which had eluded all previous inventers including Euler. The first
few pairs of amicable numbers are (220, 284), (1184, 1210), (2620, 2924), (5020,
5564) and (6232, 6368). As of 1946 there were 390 known pairs of amicable
numbers. With the advent of computers there are more than 12,000,000 pairs of
amicable numbers in 2007.
Amicable numbers are also featured in the fiction novels of
which mention must be made of the novel ‘The Professor’s Beloved Equation’ by
Yoko Ogawa and the Japanese film based on it and the novel ‘The Stranger House’
by Reginald Hall.
While these are fictions there is a rather sweet story worth
mentioning titled ‘Mathematical Aphrodisiac’ written by Alex Galt for a
compilation of short stories called “True Tales of American Life’ by Paul
Auster. The story just goes to show how mathematics can play a part in all
facets of life. The story is about the days when Alex Galt and John used to
break up all the time and they decided to see each other only casually. Dates
were okay but no more than once a week. They were going to have separate lives,
getting together occasionally when the spirit moved them, but without worrying about
any commitment.
One day they were sitting together on the floor of John’s one
room apartment. John was knitting sweater and Alex was reading Fermat’s last
theorem in a Mathematics book. And every
now and then she interrupted his knitting to read him passages from the book.
‘Did you ever hear of amicable numbers? They are like perfect
numbers but instead of being the sum of their own divisors, they are the sum of
each other divisors. In the Middle Ages people used to carve amicable numbers
into pieces of fruit. They would eat the first piece themselves and then feed
the other one to their lovers. It was a mathematical aphrodisiac. I love
that-Mathematical Aphrodisiac’, she said.
John showed little interest. He does not like mathematics and
not like Alex does. That was one of the reasons for them to be casual in their
relations.
Christmas fell during this period. Alex hates to shop. Since
Alex and John were too casual for present Alex was too glad to cross John off
from her shopping list. While Alex was shopping for her grandmother, however
she saw a cryptic crossword puzzle book and brought it for John. They had
always worked on the cryptic crossword puzzles published at the back page of a
leading daily Newspaper. Since it cost
only $ 5 she figured that she could give it to him.
When Christmas rolled over Alex had given the book to John
unwrapped and very casually. He did not give her any present. She was not
surprised, but her feelings were hurt, even though she was not supposed to
care. The next day, John invited her for dinner to his apartment.
“I have your Christmas present’, he said and added ‘sorry, it
is late”.
He handed her an
awkwardly wrapped bundle. When Alex pulled it open, a rectangle of half knit
fabric fell on her lap. She picked it up and looked at it, completely confused.
One side had the number 124,155 knitted into it and the other
side had 100,485 knitted. When Alex looked up at John, he was barely able to
contain his excitement any longer.
“They are amicable pair of numbers”, he said. “I wrote a
computer program and let it run for 12 hours. These were the biggest ones I
found and then I double knit them in the cloth which is a pot holder. I could
not give it to you last night but still I have not figured out how to cast off.
It is a kind of geeky, but I thought you might like it”, he added.
After that Christmas, they were not casual and did many
things together. They felt that the ‘Ancient Mathematical Aphrodisiac’ worked.
But for Romance and Mathematics the world would not have,
perhaps progressed.
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