Last week, a very big number — over 23 million digits long — became the “largest known prime number”.
The number, 2 77,232,917 -1, was discovered using a software called GIMPS, which allows volunteers to search for Mersenne prime numbers (more on that below). Jonathan Pace, a volunteer from Tennessee, made the discovery on December 26, and it was further confirmed using four different programs on four different pieces of hardware.
In case you want to look at the 23-million-digit number, here is the link: http://www.mersenne.org/primes/digits/M77232917.zip
What are prime numbers and why are they important?
A prime number is a number that can only be divided by itself and by 1. For example: 2, 3, 5, 7, 11, and so on.
British mathematician Marcus du Sautoy, in his book The Music of the Primes, writes, “Prime numbers are the very atoms of arithmetic. The prime numbers 2, 3 and 5 are the hydrogen, helium and lithium in the mathematician's laboratory. Mastering these building blocks offers the...hope of discovering new ways...through the vast complexities of the mathematical world.”
Dr. Baskar Balasubramanyam, Assistant Professor at the Department of Mathematics, IISER Pune, explained in detail about the new discovery in an e-mail to The Hindu :
Why is the new number called a Mersenne prime number?
Mersenne prime is a prime number of the form 2 n -1. For example, 7 = 2 3 -1 and is a prime, so it is a Mersenne prime.
One the other hand 11 is a prime, but it is not of the form 2 n -1. So it is not a Mersenne prime. Not all numbers of the form 2 n -1 are primes either. For example, 2 4 -1 = 15 is not a prime.
The GIMPS project looks at such numbers to figure out which of them are going to be primes.
So, through this software can we find bigger prime numbers?
One of the oldest theorems in mathematics (the Euclid theorem) says that there are infinitely many primes. So we are going to find larger and larger primes.
For number theorists, it is also important to understand if there are infinitely many primes that fit a particular pattern. For example, are there infinitely many primes of the form 4n+1? The answer is yes.
We still don't know if there are infinitely many Mersenne primes. Another 'family' that is of much interest are the Twin Primes (primes that are separated by 2 like 11 and 13).
Can you tell me about the applications of prime numbers?
One of the major applications of primality testing (testing whether a number is prime) is in cryptography (Cryptography, which is derived from the Greek word for the study of secret messaging, involves sharing information via secret codes).
This is based on the following principle: multiplying two numbers is easy, factoring a number is hard. For cryptographic applications, we need a number N that is a product of two primes p and q (N = pq). The value of N is public information, but it is very difficult to find p and q just by knowing the value of N — there are lots of possibilities for p and q.
Our credit cards, cell phones, all depend on cryptography.
