Appealing Numbers

The ancient Greeks, especially the Pythagoreans, were fascinated by whole numbers. They defined as “perfect” numbers those equal to the sum of their parts (or proper divisors, including 1). For example, 6 is the smallest perfect number-the sum of its three proper divisors: 1, 2, and 3. The next perfect number is 28, which is the sum of 1, 2, 4, 7, and 14.

The Pythagoreans were also interested in what we now call amicable numbers–pairs in which each number is the sum of the proper divisors of the other. The smallest such pair is 220 and 284. The number 220 is evenly divisible by 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284; and 284 is evenly divisible by 1, 2, 4, 71, and 142, which add up to 220. The Pythagorean brotherhood regarded 220 and 284 as numerical symbols of friendship.

About 1,500 years later, in the ninth century A.D., Arab scholar Thabit ibn Qurra (826-901) discovered a remarkable formula for generating amicable numbers: If n is a positive integer such that the numbers 3 x 2ⁿ – 1, 3 x 2^{n + 1} – 1, and 3² x 2^{2n + 1} – 1 are all prime numbers (evenly divisible only by themselves and 1), then 2^{n + 1}(3 x 2ⁿ – 1)(3 x 2^{n + 1} – 1) and 2^{n + 1}(3² x 2^{2n + 1} – 1) form an amicable pair. When n equals 1, you get the Pythagorean pair 220 and 284.

A few centuries later, in a letter to Marin Mersenne (1588-1648), Pierre de Fermat (1601-1665) revealed that he had found a second pair of amicable numbers (17,296 and 18,416) using a formula virtually identical to that proposed by Thabit. Using the same approach, René Descartes (1596-1650) discovered a third pair: 9,363,584 and 9,437,056.

Over the years, mathematicians have found additional formulas for amicable numbers, but no one has yet found a single formula or method that generates all possible amicable pairs.

Interestingly, despite the success of such mathematicians as Leonhard Euler (1707-1783) and Adrien Marie Legendre (1752-1833) in finding additional amicable pairs, the second-lowest pair was not found until 1867. The pair, consisting of the numbers 1,184 and 1,210, was discovered by a 16-year-old Italian youth, probably using trial and error. In recent times, extensive computer searches have uncovered many more amicable pairs.

Mathematicians now know of more than 1 million amicable pairs. “It still hasn’t been proved that the number of amicable pairs is infinite,” says Jan Munch Pedersen of Vejle Business College in Denmark, who maintains a database listing all known pairs (1,118,555 as of Jan. 30, 2001). “However, the explosive growth of pairs in the last few years leaves no doubt that this conjecture is true,” he adds.

Another unsolved puzzle concerns the parity of amicable pairs. In every known pair, both numbers are even or both are odd. However, no one has yet proved that no pair exists in which one number is odd and the other is even.

It’s amazing how much effort has gone into tracking down amicable numbers, which have practically no application in mathematics. They have a curious appeal that has endured for millennia.