Connecting Mersenne Primes to Perfect
Numbers
Even before Mersenne primes had a name,
mathematicians worked with them. Euclid, one of the alltime great math wizards, explored them around 300 B.C. In
his classic book The Elements, he proved something amazing: every Mersenne prime, multiplied by the
largest power of two less than itself, produces a perfect number.
Perfect
numbers
What is a "perfect number"? I'm glad you
asked!
A number is perfect if it equals the sum of its "proper factors" (that is, all of its factors
except itself). For example, the factors of 6 (other than 6) are 1, 2, and 3; since 1 + 2 + 3 = 6, that makes
6 a perfect number. The next one is 28 (1 + 2 + 4 + 7 + 14 = 28).
(Just as an aside, if the sum of a number's proper
factors are less than the number, it's a deficient number. If the sum is greater than the number, it's an
abundant number.)
Making perfect numbers from Mersenne
primes
Euclid's use of Mersenne primes (which weren't
called that yet, since Mersenne wasn't born until nearly 2,000 years later) was ingenious. Here's an example of the
rule cited above:
7 is a Mersenne prime. Multiply it by the largest
power of 2 that's less than 7, which is 4: 7 x 4 = 28, a perfect number.
The next Mersenne prime is 31. Multiply it by the
previous power of 2, which is 16, and you get 496. Trust me, that's a perfect number too!
Finding — and proving — that relationship is
a brilliant performance by someone working thousands of years ago, don't you think?
