Honestly, this is pretty much a "because we can" thing, other sibling replies to the contrary. It is cool and fun, and the resources dedicated to this are a pittance compared to, oh, say, the amount of computational resources applied to playing pretty 3D games.

> the amount of computational resources applied to playing pretty 3D games.

The cool thing is that those pretty 3D games are one of the reasons why we have those nifty computational resources such as http://www.nvidia.com/object/tesla-servers.html.

I think that the resources required deserve to be considered. If these computers are running 24 hours per day, and they require an extra 100 watts to look for primes (over their idle power consumption), then each computer is using the equivalent of 10 kg of coal per month.

In some cases, they're using renewable energy, and in some cases, they're offsetting heating costs. But the important thing is that the people installing the distributed computing software are not the people looking at the electricity bill. And that kind of breaks the whole economics of it.

Yes, but large numbers tend to just blow people's minds. Compare it to, say, power lossage due to people leaving their light bulbs on too long, or using incandescent bulbs instead of something more efficient.

It's noise. Nothing more.

It's used in field of cryptography. http://www.claymath.org/posters/primes/ http://primes.utm.edu/notes/faq/why.html

Not Mersenne primes though. There are only a few of them, and it would be trivial to check if any of them are used as primes. The whole point of encryption schemes that use prime numbers is that the attacker doesn't know which primes are used.

Large primes are used in cryptographic algorithms as well. http://math.stackexchange.com/a/43120

Not Mersenne Primes, because they have a special structure that can create weaknesses.

While Mersenne primes make breaking discrete logs easier over prime fields, they're pretty OK to use as underlying field on elliptic curve groups (cf. p521).

And such encryption relies on the primes that are used remaining secret.

Using any of the 'well known' primes would be a bad idea as it is easy to go through the 'well known' primes and test to see if they were used as a basis for the encryption.