EncryptionWhy do we need to know about prime numbers with millions of digits?
Prime numbers are more than just numbers that can only be divided by themselves and one. They are a mathematical mystery, the secrets of which mathematicians have been trying to uncover ever since Euclid proved that they have no end. An ongoing project – the Great Internet Mersenne Prime Search – which aims to discover more and more primes of a particularly rare kind, has recently resulted in the discovery of the largest prime number known to date. Stretching to 23,249,425 digits, it is so large that it would easily fill 9,000 book pages. You may be wondering, if the number stretches to more than 23m digits, why we need to know about it? We need to know about the properties of different numbers so that we can not only keep developing the technology we rely on, but also keep it secure. But whether or not huge prime numbers, such as the 50th known Mersenne prime with its millions of digits, will ever be found useful is 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.
Prime numbers are more than just numbers that can only be divided by themselves and one. They are a mathematical mystery, the secrets of which mathematicians have been trying to uncover ever since Euclid proved that they have no end.
An ongoing project – the Great Internet Mersenne Prime Search – which aims to discover more and more primes of a particularly rare kind, has recently resulted in the discovery of the largest prime number known to date. Stretching to 23,249,425 digits, it is so large that it would easily fill 9,000 book pages. By comparison, the number of atoms in the entire observable universe is estimated to have no more than 100 digits.
The number, simply written as 2⁷⁷²³²⁹¹⁷-1 (two to the power of 77,232,917, minus one) was found by a volunteer who had dedicated 14 years of computing time to the endeavor.
You may be wondering, if the number stretches to more than 23m digits, why we need to know about it? Surely the most important numbers are the ones that we can use to quantify our world? That’s not the case. We need to know about the properties of different numbers so that we can not only keep developing the technology we rely on, but also keep it secure.
Secrecy with prime numbers
One of the most widely used applications of prime numbers in computing is the RSA encryption system. In 1978, Ron Rivest, Adi Shamir and Leonard Adleman combined some simple, known facts about numbers to create RSA. The system they developed allows for the secure transmission of information – such as credit card numbers – online.
The first ingredient required for the algorithm are two large prime numbers. The larger the numbers, the safer the encryption. The counting numbers one, two, three, four, and so on – also called the natural numbers – are, obviously, extremely useful here. But the prime numbers are the building blocks of all natural numbers and so even more important.
Take the number 70 for example. Division shows that it is the product of two and 35. Further, 35 is the product of five and seven. So 70 is the product of three smaller numbers: two, five, and seven. This is the end of the road for 70, since none of these can be further broken down. We have found the primal components that make up 70, giving its prime factorization.
