16. Interesting Numbers


inscribed with these numbers on the grounds that they would promote love. An Arabian mathematician claimed that people could write one of the pair of numbers on one fruit and eat it, then write the second number on another fruit and give it to a lover as a mathematical aphrodisiac!


There was only one pair discovered until Pierre de Fermat discovered the pair 17,296 and 18,416 in 1636. Descartes discovered the pair 9,363,584 and 9,437,056 in 1638. A 16-year-old Italian, Nicolo Paganini, found 1,184 and 1,210 in 1867. Computers can be programmed to find larger ones now!


Remember:


When dealing with friendly and sociable numbers, the number itself is not considered a divisor.


Example 1: Verify that 1,184 and 1,210 are, in fact, friendly using a calculator. Start by dividing 1,184 by 2, then by 3 etc. The divisors of 1,184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592. Sum = 1,210.


The divisors of 1,210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605. Sum = 1,184.


Example 2:


Divisors of 10 = 1, 2, 5, 11 = 1


Prove there are no friendly pairs between 10 and 20. sum = 8 sum = 1


12 = 1, 2, 3, 4, 6 sum = 16 13 = 1


14 = 1, 2, 7 15 = 1, 3, 5 16 = 1, 2, 4, 8 17 = 1


sum = 1 sum = 10 sum = 9 sum = 15 sum = 1


18 = 1, 2, 3, 6, 9 sum = 21 20 = 1, 2, 4, 5, 10 sum = 22


PROJECT 16.8 Find the Friendly Pair


Pythagoras and his followers found one friendly pair over 2,000 years ago. Both numbers are between 200 and 300. Work in small groups of 2 - 4 to try to find that pair. You will experience, in a small way, what it is like to discover something. (See hint on page 155.)


} 149


There are no friendly pairs here. The sum of 12’s divisors = 16 But the sum of 16’s divisors ≠ 12 The sum of 14’s divisors = 10 But the sum of 10’s divisors ≠ 14 The sum of 16’s divisors = 15 But the sum of 15 divisors ≠16


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