Primality of primes

Are there infinite primes?
Euclid proved that there are infinite number of prime numbers. Simplest form of the proof is as follows. Assume there are N prime numbers. If you multiply all those numbers and add 1, you get Euclid's number. This number is not divisible by any of the primes. Therefore it is either a prime or divisible by one of the primes which was not included in the set N. It concludes that there must be N + 1 primes. Hence resulting in infinite number of primes.

How are they distributed?
The next interesting question is how these primes are distributed. UTM has more details on the distribution of primes.

What is the current largest prime known?
The largest known prime, as of December 2005, is 2^(30402457) − 1 (this number is 9,152,052 digits long); it is the 43rd known Mersenne prime. M30402457 was found on December 15, 2005 by Curtis Cooper and Steven Boone, professors at Central Missouri State University and members of a collaborative effort known as GIMPS. Before finding the prime, Cooper and Boone ran the GIMPS program on a peak of 700 CMSU computers for 9 years. (Source: wikipedia)

Where are the prime numbers used?
They are widely used in cryptography, pseudo random number generators and hash tables.

Primality proving algorithms:
Three indian scientists Manindra Agrawal, Neeraj Kayal and Nitin Saxena of IIT Kanpur came up with an algorithm known as AKS primality test to prove the primality of a number. It is the first algorithm which guarantees the distinction between a prime and composite number. It runs in polynomial time.

How would you send the treasure box safely?

Assume you have a friend in far far land. You want to send your friend a treasure box, which can be locked with multiple locks. Assume you have unlimited number of locks and corresponding keys to those locks. However your friend does not have those keys. Obviously you cannot send the key along with the treasure box. Sending the key separately is not an option. How would you send the treasure box safely?

How many lockers are open?

In a strange little town there is a school with 1000 students. Each student has a locker. The first student enters the locker room and opens all the lockers. The second closes all the even numbered lockers 2, 4, 6,...etc. The third student reverses the locker position (open locked doors and close opened doors) of locksers 3, 6, 9...etc. Similarly fourth student reverses the locker position of 4, 8, 12..etc. All the 1000 students follow the same procedure. The question is, after the 1000th student how many doors are still open?

Transparent ICs are here finally

Oregon State University researchers have successfully fabricated the world's first completely transparent IC. DeviceForge has a detailed report on this. Researches claim the technology could improve solar cells and could also be used in automobile windshields, cell phones, etc. However the practicality of the product is still elusive.

Was Einstein wrong?

We all have heard the story of identical twins, which was used to explain the Einstein's theory of relativity. However a recent research indicates that cosmic radiation might make you prematurely old due to telomere loss. NASA has a detailed report on this on their website.

Bypass mandatory registrations

Tired of filling out the registration forms online? Here is a great website where you can get temporary login names and passwords just to do that. All you need to do is copy paste the URL of the website requiring subscription, Bug Me Not gives you the login name and password. Websites like Newyork Times, Washington Post etc can be accessed without subscription. Have fun...!!!

History of photography

• Aristotle made a casual reference to Camera Obscura (meaning dark chamber) in 330BC while showing sun making a circular image when it shines through a square hole.
• Photography is greek work for light and writing, first used by Johann von Maedler in 1839.
• First commercial camera was made by Alphonse Giroux of Paris in 1839. They made around 250 cameras, each costing $50.
• Kodak marketed Brownie in 1900. It costed $1.
• The first Polaroid film was produced by Polaroid corporation in 1947.
• Vanguard 2 spacecraft took the first photo of the earth in 1959.
• In 1991, Kodak came up with the first digital camera, a Nikon F-3 camera equipped by Kodad with a 1.3 megapixel sensor.
  

What is wrong with this math proof?

         Let x = y
      xx = xy
 xx - yy = xy - yy
(x + y)(x - y) = y(x - y)
   x + y = y
   x + x = x
      2x = x
       2 = 1

Answer

Googol trivia

Ever wonder how Google got its name? It stemmed from the word GOOGOL. Googol is 1 followed by 100 zeros which was coined back in 1938, when it was the largest number. Currently googolplex is the largest number which is 1 followed by 1 googol zeros. Naming large numbers has existed right from Archimedes time. He used a unit called myriad myriad first number (1 followed by 8 zeros), myriad myriad second numbers (1 followed by 16 zeros) and so on to estimate the number of sand grains needed to fill in the known universe. His estimation was One thousand myriad myriad eighth numbers. However today we know that this much sand would fill in Milky Way Galaxybut not the entire universe.

• Googol is much larger than the number of particles in the known universe.
  
• The largest number that can be represented in a typical pocket calculator is 0.999999 googol.

Math puzzle that could land you a job at Google

Find the first 10 digits prime found in consecutive digits of e (base of natural log) which would lead you to a website for some more puzzles. If you can solve those puzzles, you could get an interview from Google. Google posted this puzzle in an attempt to find better engineers. This puzzle appeared on a hoarding in Silicon Valley. Here is an attempt by Marcus to solve this puzzle.

Fast string matching algorithms

String matching algorithms constitute a major chunk of computation time in general. Amir et al have devised a new algorithm which finds all locations where the target string differs from the source string by at most k mismatches in O(n sqrt(k log(k))).

Searching scientific material

A team lead by Robert Miner from Design Science, Inc is working on a unique project of searching mathematical content on the web. Traditional text search engines like Google, Yahoo, MSN could use this add-on feature to extend their searching capability. The current version is powered by open source Apache Lucene a modified vector space model based text search engine API. More information on this project can be found at the company's website.