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Playing with M(3326400)

Previously, I provided a review of Carlos Rivera and his life as a "prime hunter." This week, the focus shifts to a search for a special prime...and how you and your students can be part of the "hunt."

Mersenne numbers are numbers of the form 2n-1. They are named for the French mathematician Marin Mersenne (1588-1648), who actually had very little to do with them.

Prime factors of Mersenne numbers often are useful to mathematicians for cryptography, coding theory, and generating random numbers. Thus, the hunt is on....with the current search being for the prime factors of the Mersenne number 23326400-1 (or what the in-group calls M(3326400).

ElevenSmooth is a web site devoted to this search for M(3326400). Why this peculiar name? In number theory circles, a number is "smooth" if it factors into small primes. Specifically, a number is "11-smooth" if none of its prime factors exceed 11. Some known examples of 11-smooth numbers are 33, 77, 121, 242, and 231. For the current Mersenne hunt, 3326400 = 26 x 33 x 52 x 7 x 11, so 3326400 is 11-smooth. Makes sense!

The object of the web site ElevenSmooth is to get others (with their computers) to join in the search for M(3326400). That is, using what is called the distributed computing model, each participant joins in and uses special software (free) to search an assigned range of numbers. BEWARE: the task is not trivial, as the full decimal expansion of 23,326,400-1 has 1,001,347 digits.

The project ElevenSmooth is similar to Seventeen or Bust, a distributed computing project to prove the Sierpinski Conjecture. To refresh your memory, the Sierpinski conjecture states that the lowest Sierpinski number is 78557, where a whole number k is a Sierpinski number if there are no primes of the form k2n+1 for any positive number n (for k < 2n).

So, why not check the ElevenSmooth site out....or even the more general site The Great Internet Mersenne Prime Search? See if you and some of your students can join in, using your computer at times when you are not using it. It is a chance to be part of a current mathematics research project (via exhaustion!). And, you perhaps may be the ones who make the next big discovery.