OPTIMUS PRIME(2⁷⁷²³²⁹¹⁷-1), THE LARGEST KNOWN PRIME NUMBER.

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OVERVIEW

The largest known prime number 2⁷⁷²³²⁹¹⁷-1 (two to the power of 77,232,917 minus one) recently made its debut into the world of mathematics on December 26, 2017. This announcement was made by J. Pace, G. Woltman, S. Kurowski, and their co-authors. You might be wondering why one would care about prime numbers with over a million digits, I am quite certain you will appreciate them by the end of this post.

PRIME NUMBERS

Prime numbers are natural numbers which are greater than one and is not a product of numbers less than itself. Numbers that are not prime numbers are known as composite numbers. examples of prime numbers include 1, 5, 7 and even our latest prime number 2⁷⁷²³²⁹¹⁷-1. Prime numbers are a mathematical mystery which mathematicians have been trying to solve for years since Euclid proved their existence to infinity around 300BC . They are interesting entities that are very important in diverse branches of maths and are also essential to a number of real world applications.
There is no known mathematical formula that separates prime numbers from composite numbers but it is possible to statistically model the distribution of prime numbers within natural numbers. The prime number theorem is a result of such endeavour which states that "the probability of a randomly chosen number being prime is inversely proportional to its logarithm''. There are many theorems and questions as regards prime numbers all of which we will not be covering in the course of this post.

Primality is the property of being rime. As integers get large, the distance between prime numbers gets larger. This means that as the numbers ascend, the number of prime numbers reduce.

There are various methods of testing the primality of a number and they include the trial division tests, the Miller–Rabin primality test, and the AKS primality test, the above methods are slow but fast methods are available for numbers of special forms, such as Mersenne numbers which our newly discovered prime number belongs to.

MERSENNE PRIME

Mersenne prime numbers are prime numbers of the form M_n = 2ⁿ − 1, they are named after a French, Marin Mersenne who studied them in the early 17th century. It is worthy of note that not all Mersenne numbers are prime numbers. If n is composite, it implies that M_n is a composite number. Due to this, for M_n to be prime, n must be a prime number of one and no zeros.

OPTIMUS PRIME

As a big fan of cool names I decided to name our largest prime number 2⁷⁷²³²⁹¹⁷-1 optimus prime curled from the Iconic movie The Transformers. It is about a million digits more than the previous record holder. It is 23,249,425 digits long and can easily fill 9,000 book pages

HOW IT WAS DISCOVERED

The Great Internet Mersenne Prime Search(GIMPS) is an ongoing project aimed at discovering more and more Mersinne primes. This large prime was discovered by a volunteer, Jonathan Pace who had dedicated 14 years to the project. This is the 50th Mersenne prime to be discovered and just in case you think you have what it takes to find the 51st there is a $3,000 price to be claimed, visit https://www.mersenne.org/ for more information.

The discovery of this prime number is not fueld by it's possible application but by the ongoing quest by mathematicians to decipher the mystery of prime numbers. Godfrey Harold Hardy, a celebrated British mathematician said that "Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics". Notwithstanding, large prime numbers have an amazing applications in today's world which we will be discussing immediately.

ENCRYPTION WITH PRIME NUMBERS

The need to send information over the internet securely cannot be over emphasised, from texting a loved one to sending sensitive information through emails all need to be done in secrecy so as to prevent such information from getting into undesired hands and the only way to do this is through data encryption.

In 1978, Ron Rivest, Adi Shamir and Leonard Adleman invented the RSA encryption system which is named after them. This system allows for the secure transfer of information such as credit card numbers online. The RSA system is not only synonymous with public key cryptography, it is certainly the most widely used.

The first ingredient required for the RSA algorithm are two large primes. Larger prime numbers imply safer encryption. The counting numbers one, two, three, and so on are obviously, extremely useful here. But prime numbers are the bed rock of all natural numbers (all integers are made up of primes except 0 and 1) and so even more important. The strength of this system relies upon the difficulty of finding prime factorisation.

Let us take a simple illustration to boost your understanding, consider the number 130, division tells us that it is a product of 2 and 65, further breakdown tells us that 65 is a product of 5 and 13, since none of this numbers can be broken down into smaller products, we can say that we have found the primal factorisation of 130 with 2, 5 and 13 as its prime components. Factorizing numbers like 130 might be easy given that we where able to find it's primal components but that wasn't very fast. Imagine a scenario where those primal components were very large integers. I guess I will be leaving that imagination to you.

Lets apply the simple illustration above to explaining how the RSA encryption system works. Say you have a public key which is made up of the product of two large prime numbers used to encrypt your information and a secret key consisting of those two prime numbers used to decrypt the information. You are at liberty to make the public key public, allowing others to send encrypted data to you. Since you and you alone know the prime factors, you can decrypt the information while every other person will have to factor the product to get your original prime numbers which is extremely difficult. The sun would have exploded before such decryption would have been completed. This is the logic of the RSA encryption system which has held its ground and stood the test of time and scrutiny.

WORTHY NOTES

• If at any time in the future, a mathematician invents an algorithm that can factor large prime numbers, the RSA system will be rendered useless and another use for large primes will have to be discovered.

• Optimus prime is so large that at present, no computing technology is capable of using it for cryptographic safety, but as computers get more powerful and knowledge continues to increase, one can only say that nothing is impossible and just maybe optimus prime might answer the call to service someday.

SOURCES

www.mersenne.org

Stallings, W., Cryptography and Network Security: Principles and Practice,
Second Edition, New Jersey: Prentice-Hall, 1999

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