Two new Mersenne primes: 243,112,609-1 and 237,156,667-1. The former is now the largest prime number known. Interestingly the larger was discovered before the former, thus winning $100,000 from the EFF for Edson Smith who installed the software which identified this Mersenne prime on a UCLA computer. The $100K prize was for the first 10 million digit prime. The next prize is $150K for a 100 million digit prime number. Pretty amazing that two 10 million digit Mersenne's were discovered within weeks of each other.
Mersenne primes are prime numbers which are a power of two minus one, i.e. of the form 2n-1 where n is an integer. Mersenne primes are named after Marion Mersenne, a french dude sometimes known as the father of acoustics. He made a list of the known Mersenne's up to 257 in the exponent. Who said list makers don't get credit.
There are thought to be an infinite number of Mersenne primes, but no one knows how to prove this. Now days, Mersenne primes are sought by computer as there are good algorithms for testing their primality. The largest non-Mersenne prime known right now is 19,249 (213,018,586) + 1. Edouard Lucas spent 19 years testing whether the Mersenne number 2127-1 was prime. Luckily for him it was. Wouldn't that have sucked if after eighteen years he had found that it wasn't prime?
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I think you need to fix the parentheses on the expression for the largest non-Mersenne prime (otherwise, it is quite composite).
In addition to the usual "largest known perfect number" consequence, these new primes lower the exponent for the best known locally decodable codes to below O(n0.0000000232).
That should say the complexity of the best such codes, not the exponent.
So... the product of these two new Mersenne primes would be the largest known squarefree semiprime, right?
Not only did they discover, the largest prime number, but this was also the number of points that their football team gave up in the last game.