The Largest Known Prime Number

The first mention of prime numbers was by Pythagoras.

In October 2013, a new prime number was found. It is 2^57,885,161 − 1 (also approximately written as
5.818872662322464421751002121132323686363708 × 10^17,425,169
in scientific notation) having a whooping 17,425,170 digits! This beats the old 2008 record of the Largest known prime number, which is 2^43,112,609 - 1. It was discovered at the GIMPS (the Great Internet Mersenne Prime Search). The mathematician Euclid of Alexandria "the Father of Geometry," proved that there is no largest prime number, since there are infinitely many prime numbers.
Although all mathematicians and geniuses know this, they continue to discover the next one because it's an easy question. Is it a prime? If no, then ignore it and bring me the next number (and repeat). One of the main benefits of discovering this huge prime number is in Cryptography, where RSA Cryptosystem relies on the fact that computers were really slow when factorise a huge integer into (huge) prime factors. However, these people could not have done it without the help of some of the most important theorems - two of which are the Prime Numbers Theorem and the Fundamental Theorem of Arithmetic.

The Prime Number Theorem

For full article, please view the Prime Number Theorem page

The graph y=ln(x)

The Prime Number Theorem describes how the percentage of numbers that are prime numbers changes as the numbers get higher.  If a number k is randomly chosen from a range of 0 to n, the probability that it is prime will approach $\frac {1}{ln(k)}$ as n grows to infinity. In other words, prime numbers become less common as the sample size increases.

• Infinitely Many Primes
• Colollary

As we mention in the beginning of this page, there are infinitely many prime numbers. Here is one of the proofs:

• Assume there's a largest prime number, called $P$.
• Multiply all the prime numbers up to $P$ (which is going to be $2 \times 3 \times 5 \times 7 \times 11 \times ... \times P$) and name it $M$.
• Note that $M + 1$ is a prime
  • The Next multiple of 2 after $M$ is $M + 2$
  • The Next multiple of 3 after $M$ is $M + 3$
  • The Next multiple of 5 after $M$ is $M + 5$
  • The Next multiple of 7 after $M$ is $M + 7$
  • The Next multiple of 11 after $M$ is $M + 11$
  • and so on...
• This shows that there is a larger prime number after $P$, which is a largest prime number, according to our assumption at the beginning
• That means our assumption at the start is false: the Largest prime number does NOT exist! there's infinite of them.

Corollary 1

The graph of y=log(x). Note how this is similar to the previous graph.

$\pi(x)$ is approximately equal to $\frac {x}{log(x)}$

Corollary 2

The k^th prime number is approximately equal to $k \times log(k)$. However, keep in mind that this is still an approximation; there is no real way of determining the exact number.

Corollary 3

The probability of a random integer k being prime is approximately $\frac {1}{log(k)}$. This is true since that log base e of k is equal to $ln(k)$. Note that their graphs are similar, as shown with the graphs of $y=ln(x)$ and $y=log(x)$ as shown in the top-right pictures.

The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic, or the unique prime factorization theorem, states that an integer k (such that $0 < k < \infty$) is either a prime or can be composed into other prime numbers (hence the name composite numbers). In other words, each integer has one and only one unique Prime factorization (in the set of real numbers), no matter how it is done.

For example: the prime factorization of 1000 is 2³ x 5³. No matter how 1000 is broken up, it will be 2³ x 5³ when expressed as a prime factorization, and you can't achieve the same number with different prime combinations.

Proof

There are many proofs for this theorem, but three are using the existence theorem, uniqueness theorem, and the proof of the uniqueness theorem.

TOP 20 Largest Known Prime Numbers

As of 2013, here are the top ten largest known prime numbers.

Rank	Prime number	Found by	Found date	Number of digits
1st	257,885,161 − 1	GIMPS	25th January 2013	17,425,170
2nd	243,112,609 − 1	GIMPS	23th August 2008	12,978,189
3rd	242,643,801 − 1	GIMPS	12th April 2009	12,837,064
4th	237,156,667 − 1	GIMPS	6th September 2008	11,185,272
5th	232,582,657 − 1	GIMPS	4th September 2006	9,808,358
6th	230,402,457 − 1	GIMPS	15th December 2005	9,152,052
7th	225,964,951 − 1	GIMPS	18th February 2005	7,816,230
8th	224,036,583 − 1	GIMPS	15th May 2004	7,235,733
9th	220,996,011 − 1	GIMPS	17th November 2003	6320,430
10th	213,466,917 − 1	GIMPS	14th November 2001	4,053,946

11th 2^~10,000,000 glmps ??th ?? 19??

Trivia

• the EFF (the Electronic Frontier Foundation) gives out prizes for new prime numbers discovered in the world. In 1999, this foundation granted a $50,000 prize. This theorem, indirectly. may help mathematicians and other geniuses to find new ones.  Although it is not accurate, it gives a ballpark of numbers rather than shooting at the dark.
  • 8If the number of digits in the prime number exceeds another certain amount of digits, the foundation will give out another prize.
• If you pay attention to the largest known prime number table above, all of the entries can be expressed in form 2ⁿ - 1, which is Mersenne Prime general form.

Important Pages
Divisibility and Sieves	Divisibility Rules (by rule) | (by list) | Sieve of Eratosthenes
Proofs and Theorems	Euclid's Proof | Goldbach's Conjecture | Prime Number Theorem | Integer
Prime Numbers	Timeline of the History of Prime Numbers | The Largest Known Prime Number
Help	Getting Started
Different Bases	What Digit a Prime Number in a Base Can End In