What is the greatest prime number of them all?

One of the greatest mathematical challenges and brain twisters of our time is the search for the greatest prime.

Euclid proved that there are an infinite number of prime numbers in his book, The Elements. One might think that people would be satisfied with knowing that the set of prime numbers is not finite and therefore cease their search. However, the hunt lives on today.

The Great Internet Mersenne Prime Search (GIMPS) is an organization seeking out the next larger prime number. To this end, the individuals involved use powerful computers and efficient algorithms to try and calculate it. When a new prime is discovered as a result of the stringent, tough, and interminable process, thorough testing must be done by several sources in order to make sure that it wasn't a fluke find. For instance, there could be a memory leak or some other issue on the computer involved that causes a miscalculation. If something like this happens, it could cause the outputted number to be incorrect. The people who are responsible for contributing to this effort need to be sure that they aren't making non-prime numbers famous.

The equation/proof commonly used in the calculation of Mersenne primes is called the Lucas-Lehmer Test and can be expressed as follows:

For p an odd prime, the Mersenne number 2p-1 is prime if and only if 2p-1 divides S(p-1) where S(n+1) = S(n)2-2, and S(1) = 4

The GIMPS is very similar to another numbers crunching computational process that I actually mentioned in this prior post. Folding@Home hopes to crunch its way to a cure for serious illnesses and diseases.

* The Elements of Euclid
* Mersenne
* Great Internet Mersenne Prime Search

Computer Graphics

"A picture is worth a thousand words."

This idea has been circulating for just over a century. It was said at a time when newspapers were the norm, not computers. What does it mean for us in our now technologically advanced society (relative to the early twentieth century)?

We've come to a time when graphics absolutely must be used in order to tell some kind of story. But it goes much deeper than that. Take for instance, a video game. Video games have, in recent years, used special imaging in order to capture the motion of a real person and then paint over polygons with a face. However, this was limited by the power of the machinery behind it. Now we can actually create a model of a head using modern computational powers.




Our technology has surpassed barriers that we used to think permanent. We can now sculpt human heads that look quite eerily identical to a real person's.

The most important thing to remember is we were further than this just a couple of centuries ago. In the late 80s and 90s, we would use the Bresenham line algorithm in computer programs. This algorithm is  used mostly for plotters and thusly the creation of vector graphics. However, its contribution to the world of computer generated graphics shows how fast technology is moving.

* http://www.idav.ucdavis.edu/education/GraphicsNotes/Bresenhams-Algorithm.pdf

Internet Message Access Protocol and Post Office Protocol

Internet Message Access Protocol (IMAP) and Post Office Protocol (POP) are the mainstream protocols that most e-mail services use currently. Both have fairly significant differences. Why does it matter how your client is set up? See, you can setup, say, your smartphone's e-mail application to either use POP or IMAP.

IMAP almost always stores the inbox itself (consisting of all emails received) on the central e-mail server. One benefit of this service is that all mail is backed up frequently. In the case that there is some kind of large-scale crash, your data is preserved in more than one server location. Once the outage has been fixed, then the e-mail attached to your account will return to your regular inbox as per usual operation. IMAP employs the use of ciphers, specifically for MD5 and SHA hashes for connections between client and server. If a match fails authentication, then a user must intervene to provide some form of confirmation or risk the connection itself being severed and labeled as suspect. A possible issue affecting this security process is that of a rough connection. Perhaps some packets are lost in transit, thereby weakening the security on one end, which would make authentication that much more complicated.

POP works fairly differently than IMAP. E-mail is stored initially on a server and later saved relatively permanently on the computer being used. However, as a result, the messages will almost always no longer be stored on the server indefinitely unless configured in a special way. So there will be no way to resurrect lost data unless the user has been keeping track. POP uses specific commands with shared secrets for security. Secrets can cause accessing private data to become more complex.

Security-wise

* http://tools.ietf.org/html/rfc3501
* http://tools.ietf.org/html/rfc5321
* http://tools.ietf.org/html/rfc1939