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90 THE FASCINATION OF NUMBERS

Somewhat similar to perfect numbers are those known as
amicable numbers. These are pairs of numbers, each of
which is equal to the sum of all the factors of the other. Thus,
the sum of all the factors of the number 220 is equal to 284,
while the sum of all the factors of 284 is equal to 220. The
next pair of amicable numbers are 1184 and 1210, and other
examples are 17296 and 18416. The other known amicable
numbers are much greater.

In another category are the Automorphic numbers, the
special property of which is that their digits reappear as the
last digits of their squares and higher powers. Examples are
25 and 76.

Palindromic numbers are those which are the same either
read forwards or backwards. The most simple form of palin-
dromic numbers is the one containing a number of identical
digits. Some palindromic numbers are themselves derived
from others, as is shown by the following table of the squares
of numbers consisting entirely of the repetition of one digit.

11%=121
I11%2=12321
I111%2=1234321
II111%2=129454321
ITI1112=12345054321
III11112=1234567654321
ITIIII112=123456787654321
IITITIT11%2=12345678987654321

Higher squares, however, give numbers of a different pat-
tern. Another interesting series is that in which the palin-
dromes form the centres of the numbers shown above:

232 =4 X 58
23432 =4 X 5858
2345432 =4 X 586358
234565432 =4 X 58641358
23456765432 =4 X 5864191358
2345678765432 =4 x 586419691358
234567898765432 =4 X 58641974691358