GIMPS Discovers The Largest Known Prime Number Yet

The Great Internet Mersenne Prime Search (GIMPS) discovered the largest prime number yet on January 7, 2016. The number is 2^{74, 207, 281} - 1. It contains 22,338,618 digits. If you are wondering how long it is, suppose that you can write an average of 2 digits per second, you will be able to finish writing this number in about 129 days without eating, sleeping, or toilet break. Using the same rate, if you are going to write this number for 6 hours a day, then you will finish it in about 517 days (roughly one and a half years).

GIMPS has been calculating large prime numbers since 1996 and has discovered 15 of the largest known prime numbers. As of this writing, the 11 largest prime numbers are Mersenne primes. Mersenne primes are prime numbers of the form 2^n - 1 for some integer n. Three of the smallest Mersenne primes are 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31 .

Question: Are all positive integers of the form 2^n - 1 prime numbers?

For the non-math persons, prime numbers are positive integers that can only be divided by 1 and itself. For example, 5 is a prime number since you cannot divide 5 by any number except one and itself. On the other hand, 8 is not a prime number because 8 can be divided by 1, 2, 4, and 8.

It was already proven by Euclid (some 2300 years ago) that there is no largest prime number, so the search for large prime numbers will never end.

Some of the most useful application of prime numbers is cryptography, particularly internet security. It’s what makes your password safe. It is what makes shopping safe. Well, you still have to be careful though.

The Geometric Representation of Greatest Common Factor

A factor is an integer that divides another integer. The number 6 is a factor of 12 since 6 divides 12. It is easy to see that 1, 2, 3, 4, and 12 are also factors of 12. Looking at the numbers in the tables below, we can see that some numbers have only 2 factors, 1 and itself. These numbers are called prime numbers.

Two or more numbers can have common factors. For example, let us consider the factors of 12 and 18.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18  » Read more

Understanding Hilbert’s Grand Hotel Paradox

Long ago, in a land far away, there was a grand hotel where there were infinitely many rooms. This hotel was attended by a brilliant manager.

One night, a guest arrived, but  the hotel was full — each room was occupied by one guest. The newly arrived guest asked if a spare room was available. “Of course we have, we are the Infinite Grand Hotel. There is always a room for everyone,” the manager said proudly.

Now since each room was occupied by a guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room 3 to move to Room 4, and so on. Basically, he told every guest in Room n to move to Room n + 1. Since the hotel had infinitely many rooms, there was no problem in moving, there was always a room to move to. This left Room 1 vacant, and therefore, the guest was accommodated. The guest was happy. The manager was happy.  » Read more

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