Friendly Numbers  
    Pythagoras considered 220 and 284 to be friendly (or amicable). He even wrote: "[A friend] is the other I, such as are 220 and 284". Aristotle also used the notion of friendly numbers to characterize friendship (in his work "Ethics"). So what makes the numbers 220 and 284 so special? Their property is that each is equal to the sum of the other's proper divisors. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, and they sum up to 284; the proper divisors of 284 are 1, 2, 4, 71, and 142, and they sum up to 220.
 
    The pair 220, 284 soon became a symbol of friendship. In the middle ages, 220 and 284 were used by a young man to express his love to his beloved. He would prepare two metal charms with the numbers 220 and 284 engraved on them. These were then put on chains to place around their necks.
 
    Ibn Qurra Ibrahim, living in Bagdad in the 9^th century, tells about the habit of taking two fruits, cutting 220 in the one and 284 in the other, eating the first one and presenting the second one to the beloved as a mathematical aphrodisiac. He even gave an algorithm for finding candidate pairs of friendly numbers. This algorithm reads
- make four rows of numbers
- write the powers 2ⁿ in the first row, starting by n=1
- write the triple of the numbers of the first row in the second row
- write the numbers of the second row in the third row, subtracting 1 of each of them
- write the product of (the number in the second row of the column you're in) times (the left neighbour of this number) minus 1 in the fourth row, starting in the second column, of course
- if two neighbouring numbers in the third row and the number in the column of the greater one of these in the fourth row are prime, then you've found the pair of friendly numbers (a, b) by:
   1^st row....x
   2^nd row.....
   3^rd row...yz
   4^th row....t    a = x*y*z, b = x*t

    For example, consider this table for n=1,...,7:
       1  2   4   8   16   32    64   128
       3  6  12  24   48   96   192   384
       2  5  11  23   47   95   191   383
         17  71 287 1151 4607 18431 73727

    The smallest pair of friendly numbers is, following the scheme from above:
   220 = 4*5*11, 284 = 4*71
 
    It seems that when Pierre de Fermat and Marin Mersenne, who in 1636 published the pair (17296 = 16*23*47 and 18416 = 16*1151), rediscovered this algorithm. Using it, René Descartes found the third pair (9363584 = 128*191*383 and 9437056 = 128*73727). Even though this rule was published, it took until 1747 when Leonhard Euler added 30 (!) more pairs (published in his article "De numeris amicabilius"), adding 32 more pairs in 1750. Euler found his pairs not by the above algorithm, and today it is known that only these three pairs can be found by this algorithm for n<20000. However, it was not until 1867, that the second smallest pair (1184 and 1210) was discovered by Niccolò Paganini, a sixteen-year-old Italian schoolboy.
 
    By now, about 12000 pairs of friendly numbers have been discovered, but no one knows whether there are infinitely many.
 
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