An integer number is a prime number whenever it is divisible by two distinct integers, 1 and itself.
An integer is divisible by another one whenever the remainder of the Euclidean division of the former by the latter is zero: for example, 18 is divisible by 3 because 18 = 3 × 6 + 0, therefore the remainder is zero (0). Conversely, 19 is not divisible by 6 because 19 = 3 × 6 + 1, hence the remainder is 1, therefore non-zero.
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Although they have been known circa 300 BC, prime numbers remain a mystery of the modern mathematics. It has been known since Ancient history (thanks to the Greek mathematician Euclid) that there exist infinitely many prime numbers; nevertheless, nowadays, it is still difficult to verify the primality of an integer (i.e. to decide whether is a prime number), especially for (very) large integers.
Applications of prime numbers are numerous, both in mathematics and computer science, and include public key cryptography (also referred to as asymmetric cryptography), used in particular for secure payment over the Internet. In fact, the difficulty to decompose a very large number into prime factors (called the prime factorization of an integer) is the basis of the security of many aspects of our digital life (payment by credit card, security of a Web site secured by HTTPS…). This difficulty to perform a prime factorization is especially true for very large numbers, containing hundreds or thousands of digits.
The largest prime number ever exhibited was obtained on 7th January 2016. It is 274 207 281-1. If it was to be written in classical decimal notation, it would be made of… 22 millions of digits! This number is a Mersenne prime, because it is written using the form 2n-1, where n is itself a prime number. This number was obtained thanks to the distributed computer software GIMPS, the goal of which is to discover new prime numbers: this program is a distributed and collaborative software that runs on computers all over the world. GIMPS mainly relies on Lucas–Lehmer primality test for Mersenne primes.
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