The largest known prime has now been bumped up. The recently discovered prime is 2^43112609 − 1 which in base 10 has around twelve million digits. As with all the largest primes discovered for most of the last hundred years, the prime is a Mersenne prime, that is it is one less than a power of 2. As with the last few Mersenne primes discovered, this was discovered by the Great Internet Mersenne Prime Search which uses distributed computing to search for Mersenne primes. I've blogged before about Mersenne primes. For more details on why we care about Mersenne primes and their history dating back to the ancient Greeks see this post and this post.
There are reasons to suspect that if 2^n-1 is prime that n-1 should be likely to have a lot of small prime factors. In this case, 43112609 -1= 2^5 * 7 * 11 * 17497. So while some of the prime divisors look very small, it has at least one prime very large prime divisor and so doesn't seem to fit this pattern well. This is in contrast to the last discovered Mersenne prime 2^42643801 - 1. In that case one has 42643800 = 2^3 * 3^3 * 5^2 * 53 * 149. It remains to be seen if this new example is simply an outlier or if we need to reevaluate what we expect the exponents of Mersenne primes to look like.
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