MaBloWriMo 4: not all prime-index Mersenne numbers are prime

Over the past couple days we saw that if n is composite, then M_n is also composite. Equivalently, this means that if we want M_n to be prime, then at the very least n must also be prime. But at this point there is an obvious question: is M_n always prime when n is prime? Or are there some prime n for which M_n is (sadly) composite?

Well, if M_n were always prime whenever n is prime, then I wouldn’t be writing this series of blog posts! That would be a nice, simple situation with no need for any fancy tests. But let’s see what actually happens:

  • M_2 = 3 is prime.
  • M_3 = 7 is prime.
  • M_5 = 31 is prime.
  • M_7 = 127 is prime.
  • M_{11} = 2047 is… composite! 2047 = 23 \times 89.

It turns out that M_{13}, M_{17}, and M_{19} are all prime again, but then M_{23} is not (2^{23} - 1 = 8388607 = 47 \times 178481). And so on.

You get the idea: to find prime M_n, we only need to consider prime values of n. But even then we still have some checking to do, since not every prime n corresponds to a prime Mersenne number M_n. In general, checking a number to see if it is prime can take a long time; the cool thing about Mersenne numbers is that using the Lucas-Lehmer test, we can check them for primality much more quickly than other numbers of a similar size. Tomorrow we’ll recall exactly how the Lucas-Lehmer test works, and try it on some small examples.

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About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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One Response to MaBloWriMo 4: not all prime-index Mersenne numbers are prime

  1. Pingback: MaBloWriMo 5: The Lucas-Lehmer Test | The Math Less Traveled

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