45th and 46th Mersenne Primes Confirmed

kahunak writes to alert us that GIMPS has announced that the 45th and 46th Mersenne primes have been confirmed. The EFF's $100,000 award, for the first prime over 10 million digits in length, will probably be claimed. (We discussed no. 45 when it was announced.)

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Why Mersenne Primes Matter

[RAMMS EIN]

Not knowing why Mersenne primes matter, I looked it up on The Ultimate Source Of Truth. From The Fine Article:

Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether there is a largest Mersenne prime, which would mean that the set of Mersenne primes is finite. The Lenstra-Pomerance-Wagstaff conjecture asserts that, on the contrary, there are infinitely many Mersenne primes and predicts their order of growth. It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.

Mersenne primes are used in pseudorandom number generators such as Mersenne Twister and ParkMiller RNG.

Mersenne primes were considered already by Euclid, who found a connection with the perfect numbers.

Mersenne numbers are very good test cases for the special number field sieve algorithm

Out of those, I only knew about the connection with pseudorandom number generators, which I became interested in after writing my deadbeef random number generator.