Ruling out gaps in the list of Mersenne primes

The largest known primes are Mersenne primes, in part because the Lucas-Lehmer algorithm makes it more efficient to test Mersenne numbers for primality than to test arbitrary numbers. These numbers have the form 2ⁿ - 1.

The Great Internet Mersenne Prime Search (GIMPS) announced today that they have tested whether 2ⁿ - 1 is prime for all values of n less than 100,000,000 [1]. More specifically, they've tested each exponent at least once, though they'd like to test each exponent twice before giving up on it.

If we assume all of their initial calculations are correct, we can say, for example, that 2^82,589,933 - 1 is the 51st Mersenne prime. Before we had to say that it was the 51st known Mersenne prime because there could be a smaller, undiscovered Mersenne prime, in which case 2^82,589,933 - 1 might be the 52nd Mersenne prime. Mersenne primes have not always been discovered in order. For example, the 29th Mersenne prime was discovered after the 30th and 31st such numbers.

So for now, again assuming all the GIMPS calculations have been correct, we know all Mersenne primes less than 2^100,000,000, and there are 51 of them.

• Five things about Mersenne primes
• Even perfect numbers
• Fermat primes

[1] There's no need to test every value of n. We know that if 2ⁿ - 1 is prime, then n is prime. So we only need to check prime values of n.

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