|Ivars Peterson's MathLand|
January 27, 1997
Together, Euclid of Alexandria (365-300 B.C.) 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 of Muhlenberg College described this remarkable effort, which spanned nearly 20 centuries, at the Joint Mathematics Meetings in San Diego earlier this month.
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 smallest 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 2^n - 1 is prime, then [2^(n-1)](2^n - 1) is perfect.
Hence, for n = 5, 2^5 - 1 = 31, and [2^(5-1)](2^5 - 1) = 2^4(31) = 16 x 31 = 496. As a check, 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496.
Primes of the form 2^n - 1 are now known as Mersenne primes, and these numbers figure prominently in the search for the largest known prime (see Another Prime Record). Every Mersenne number uncovered automatically leads to a new perfect number. So far, 35 perfect numbers have been identified.
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 fitted 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 2,305,843,008,139,952,128. Euler himself worked out the eighth example in 1772: 2,658,455,991,569,831,744,654,692,615,953,842,176. That's pretty good at a 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 quips.
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 unique pronunciation.
Copyright © 1996 by Ivars Peterson.
Conway, John H., and Richard K. Guy. The Book of Numbers. New York: Copernicus.
Dunham, William. The Euclid-Euler theorem. Abstracts of Papers Presented to the American Mathematical Society 18(No. 1):4.
A useful summary of the link between Mersenne primes and perfect numbers can be found at http://www.utm.edu/research/primes/mersenne.shtml.
Lists of known perfect numbers are available at http://forum.swarthmore.edu/dr.math/problems/perfect.html and http://www.maths.uts.edu.au/number/perfect.html.
A page on perfect, amicable, and sociable numbers can be found at http://xraysgi.ims.uconn.edu/amicable.html.
An online, interactive edition of the Elements, can be found at http://aleph0.clarku.edu/~djoyce/mathhist/euclid.html.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.
Ivars Peterson is the mathematics and physics writer and online editor at Science News ( http://www.sciencenews.org/). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, and Fatal Defect. His current work in progress is The Jungles of Randomness (to be published in 1997 by Wiley).