Connecting Mersenne Primes to Perfect Numbers

Even before Mersenne primes had a name, mathematicians worked with them. Euclid, one of the all-time 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?