Mersenne Numbers

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Subsection 8.4.5 Mersenne Numbers

We’ll give one more example of a famous disproof by counterexample.

A Mersenne number is a number that can be expressed in the form 2n – 1, for some positive integer n. Here’s a table of the first several Mersenne numbers:

Table 8.4.1.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
2n-1	1	3	7	15	31	63	127	255	511	1023	2047	4095	8191	16383

Now let’s focus on those Mersenne numbers that happen also to be prime. We’ll call these numbers Mersenne primes. (The name comes from the 17th century monk, Marin Mersenne, who studied them.)

Exercises Exercises

1.

Here’s the Mersenne numbers table again:

Table 8.4.2.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14
2n-1	1	3	7	15	31	63	127	255	511	1023	2047	4095	8191	16383

Which of the Mersenne numbers shown in this table are prime?

Answer.

Answer: 3 7 31 127 8191.