It seems an unlikely pairing. One was the most prominent mathematician of antiquity, best known for his treatise on geometry, the Elements. The other was the most prolific mathematician in history, the man whom his eighteenth-century contemporaries called “analysis incarnate.” Together, Euclid of Alexandria (c325–c265 BC) and Leonard Euler (1707–1783), born in Switzerland and at various times resident in St. Petersburg and Berlin, collaborated on proving an interesting result in number theory–without the benefit of e-mail or time travel.
Mathematician William Dunham describes this remarkable effort, which spanned nearly 20 centuries, in his book Euler: The Master of Us All.
The story begins with the fascination that numbers held for the followers of Pythagoras in ancient Greece. Among those of special interest were the perfect numbers, which have the property that their proper divisors add up to the number itself. For example, the proper divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6.
Six is the smallest perfect number. The next highest is 28. Its divisors are 1, 2, 4, 7, and 14, so 1 + 2 + 4 + 7 + 14 = 28. Euclid also knew the next two perfect numbers: 496 and 8,128. Notice that each of the four numbers can be written as the following products: 2 x 3, 4 x 7, 16 x 31, and 64 x 127.
On the basis of this limited evidence and some careful reasoning, Euclid proved in the final proposition on number theory in the Elements (Book IX, Prop. 36) that if 2n – 1 is prime, then [2(n – 1)](2n – 1) is perfect.
Hence, for n = 5, 25 – 1 = 31, and [2(5 – 1)](25 – 1) = 24 (31) = 16 x 31 = 496. As a check, 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.
Primes of the form 2n – 1 are known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime. Every Mersenne number uncovered automatically leads to a new perfect number.
There are some other patterns that Euclid might have noticed. For example, the final digits of the four perfect numbers that he knew alternate between 6 and 8. It turns out that the final digit of an even perfect number is always 6 or 8, but the alternating pattern doesn’t hold.
The first four perfect numbers written in binary form also display a striking pattern: 110, 11100, 11111000, and 11111110000!
What Euclid didn’t answer, however, was whether every perfect number fits his formula. It’s possible that some might result from another formula.
By the time Leonard Euler came on the scene, seven perfect numbers were known. An unknown mathematician had found 33,550,336 in 1456, and Pietro Antonio Cataldi (1548–1626) had discovered two more in 1588: 8,589,869,056 and 137,438,691,328. Euler himself worked out the eighth example in 1772: 2,305,843,008,139,952,128. That’s pretty good at time when there were no calculators or Crays.
In 1747, Euler proved the partial converse of Euclid’s theorem: All even perfect numbers must have the form specified by Euclid’s formula. In other words, there is no other way to obtain perfect numbers that are even.
Euler’s proof wasn’t published until after his death. It can be said that he “perished before he published,” Dunham once quipped
The issue of odd perfect numbers remains unsettled, however. No one knows whether there are any. Mathematicians have so far proved that if an odd perfect number exists, it must have at least 300 decimal digits and must have at least 29 prime factors (not necessarily distinct).
Perhaps, in another 20 centuries, there will be a third collaborator to add to the Euclid-Euler team, when the question of odd perfect numbers is finally resolved. It would be fitting if that individual had a name that also started with “EU” and had its own distinctive pronunciation.
Originally posted: 1/25/97
Updated: 1/18/03
