Number Theory Section Summary: 10.1-2

Mersenne and Perfect Numbers

  1. Summary

    Father Marin Mersenne (1588-1648) helped drive the scientists of his day to share their results, and pass around the scientific knowledge available at that time. Yet the French priest had interests of his own, including number theory. Some of the most famous primes are graced with his name.

  2. Definitions

    perfect number: a positive integer n equal to the sum of all positive divisors, excluding n itself. [Name bestowed by the Pythagoreans.]

    e.g.

    Mersenne number: tex2html_wrap_inline226 , with tex2html_wrap_inline228 . If tex2html_wrap_inline230 is prime, then it's called a Mersenne prime

    The latest prime (as of today, a Mersenne prime) was found December 15, 2005. The 43rd Mersenne Prime, it is

    displaymath208

    and has nearly 10 million digits. There's a prize of $100,000 for the first 10 million digit prime!

    Last time I taught this course (one year ago), they'd just found the 42nd, as described in Science News: ``On Feb. 18 [2005], the computer-based Great Internet Mersenne Prime Search (GIMPS) turned up the largest known prime number, whose formula is 2 to the 25,964,951st power minus 1. The new prime is a whopping 7,816,230 digits long, making it more than half-a-million digits longer than the previous record-holder. The number would completely fill 58 issues of Science News.''). I like that analogy!

    Mersenne had conjectured that tex2html_wrap_inline232 is prime for 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, and composite for all other prime powers <257. Mersenne was wrong (five in the list are not prime, and he missed three others), as was dramatically demonstrated in 1903 by Frank Nelson Cole for the case of 67 (See story, p. 216).

  3. Theorems

    The test of a perfect number is if

    displaymath209

    Theorem 10.1: If tex2html_wrap_inline236 is prime (k>1), then tex2html_wrap_inline240 is perfect, and every even perfect number is of this form.

  4. Properties/Tricks/Hints/Etc.

    Theorem 10.2: An even perfect number n ends in the digit 6 or 8; equivalently, either tex2html_wrap_inline244 or tex2html_wrap_inline246 .

    The proof of this result relies on a lemma, which is interesting:

    Lemma: If tex2html_wrap_inline248 is prime (a>0, tex2html_wrap_inline252 , then a=2 and k is also prime.

    [As our author notes, ancients believe that tex2html_wrap_inline236 was always prime for k prime, but this is not the case: tex2html_wrap_inline262 is not prime (1536).]

    Proof:



Thu Apr 6 17:27:07 EDT 2006