The Pythagoreans studied numbers that they called amicable or friendly. A pair of numbers is friendly if each number is the sum of the proper divisors of the other (a proper divisor includes the number 1, but not the number itself). The Pythagoreans discovered that 220 and 284 are friendly. The proper divisors of 220 are {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110} and

The next pair of friendly numbers was found by Pierre de Fermat (1601–1665): 17,296 and 18,416. In 1638 the French mathematician René Descartes (1596–1650) found a third pair, and the Swiss mathematician Leonhard Euler (1707–1783) found more than 60 pairs. In 1866 a 16-year-old Italian schoolboy, Nicolo Pagonini, found another relatively small pair of friendly numbers that had been overlooked by the great mathematicians. He found the pair of numbers 1,184 and 1,210. Show that 1,184 and 1,210 are friendly.

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1+2+4+5+ 10+11+20 +22+44 +55 + 110 = 284 Also, the proper divisors of 284 are {1, 2, 4, 71, 142}, and 1+2+4+71 + 142 = 220