Why do we need to know about prime numbers with millions of digits?

Multiplying two numbers, even if very large, is perhaps tedious but a straightforward task. Finding prime factorization, on the other hand, is extremely hard, and that is precisely what the RSA system takes advantage of.

Suppose that Alice and Bob wish to communicate secretly over the internet. They require an encryption system. If they first meet in person, they can devise a method for encryption and decryption that only they will know, but if the initial communication is online, they need to first openly communicate the encryption system itself – a risky business.

However, if Alice chooses two large prime numbers, computes their product, and communicates this openly, finding out what her original prime numbers were will be a very difficult task, as only she knows the factors.

So Alice communicates her product to Bob, keeping her factors secret. Bob uses the product to encrypt his message to Alice, which can only be decrypted using the factors that she knows. If Eve is eavesdropping, she cannot decipher Bob’s message unless she acquires Alice’s factors, which were never communicated. If Eve tries to break the product down into its prime factors – even using the fastest supercomputer – no known algorithm exists that can accomplish that before the sun will explode.

The primal quest
Large prime numbers are used prominently in other cryptosystems too. The faster computers get, the larger the numbers they can crack. For modern applications, prime numbers measuring hundreds of digits suffice. These numbers are minuscule in comparison to the giant recently discovered. In fact, the new prime is so large that – at present – no conceivable technological advancement in computing speed could lead to a need to use it for cryptographic safety. It is even likely that the risks posed by the looming quantum computers wouldn’t need such monster numbers to be made safe.

It is neither safer cryptosystems nor improving computers that drove the latest Mersenne discovery, however. It is mathematicians’ need to uncover the jewels inside the chest labelled “prime numbers” that fuels the ongoing quest. This is a primal desire that starts with counting one, two, three, and drives us to the frontiers of research. The fact that online commerce has been revolutionized is almost an accident.

The celebrated British mathematician Godfrey Harold Hardy said: “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”. Whether or not huge prime numbers, such as the 50th known Mersenne prime with its millions of digits, will ever be found useful is, at least to Hardy, an irrelevant question. The merit of knowing these numbers lies in quenching the human race’s intellectual thirst that started with Euclid’s proof of the infinitude of primes and still goes on today.

Ittay Weiss is Teaching Fellow, Department of Mathematics, University of Portsmouth. This article is published courtesy of The Conversation (under Creative Commons-Attribution / No derivative).