180 7. ANCIENT NUMBER THEORYprofound insight into the divisibility properties of numbers. It is very difficult to
imagine how he could have discovered this result. A conjecture, which cannot be
summarized in a few lines, can be found in the article by Brentjes and Hogendijk
{1989).
It is not clear how many new cases can be generated from this formula, but
there definitely are some. For example, when ç = 4, we obtain the amicable pair
17,296 = 16-23-47 and 18,416 = 16 • 1151. Hogendijk (1985) gives Thabit ibn-
Qurra's proof of his criterion for amicable numbers and points out that the case
ç = 7 generates the pair 9,363,584 and 9,437,056, which first appeared in Arabic
texts of the fourteenth century.
Unlike some other number-theory problems such as the Chinese remainder the-
orem, which arose in a genuinely practical context, the theory of amicable numbers
is an offshoot of the theory of perfect numbers, which was already a completely
"useless" topic from the beginning. It did not seem useless to the people who
developed it, however. According to M. Cantor (1880, p. 631) the tenth-century
mystic al-Majriti recommended as a love potion writing the numbers on two sheets
of paper and eating the number 284, while causing the beloved to eat the num-
ber 220. He claimed to have verified the effectiveness of this charm by personal
experience! Dickson (1919, p. 39) mentions the Jewish scholar Abraham Azulai
(1570-1643), who described a work purportedly by the ninth-century commentator
Rau Nachshon, in which the gift of 220 sheep and 220 goats that Jacob sent to his
brother Esau as a peace offering (Genesis 32:14) is connected with the concept of
amicable numbers.^14 In any case, although their theory seems more complicated,
amicable numbers are easier to find than perfect numbers. Euler alone found 62
pairs of them (see Erdos and Dudley, 1983).
Another advance on the Greeks can be found in the work of Kamal al-Din
al-Farisi, a Persian mathematician who died around 1320. According to Agargun
and Fletcher (1994), he wrote the treatise Memorandum for Friends Explaining the
Proof of Amicability, whose purpose was to give a new proof of Thabit ibn-Qurra's
theorem. Proposition 1 in this work asserts the existence (but not uniqueness) of
a prime decomposition for every number. Propositions 4 and 5 assert that this
decomposition is unique, that two distinct products of primes cannot be equal.6. Japan
In 1627 Yoshida Koyu wrote a textbook of arithmetic called the Jinko-ki (Treatise
on Large and Small Numbers). This book contained a statement of what is known
in modern mathematics as the Josephus problem. The Japanese version of the
problem involves a family of 30 children choosing one of the children to inherit the
parents' property. The children are arranged in a circle and count off by tens; the
unlucky children who get the number 10 are eliminated; that is, numbers 10, 20,
and 30 drop out. The remaining 27 children then count off again. The children
originally numbered 11 and 22 will be eliminated in this round, and when the second
round of numbering is complete, the child who was first will have the number 8.
Hence the children originally numbered 3, 15, and 27 will be eliminated on the next
(^14) The peace offering was necessary because Jacob had tricked Esau out of his inheritance. But if
the gift was symbolic and associated with amicable numbers, the story seems to imply that Esau
was obligated to give Jacob 284 sheep and 284 goats. Perhaps there was an ulterior motive in the
gift!