A prime number is a natural number greater than 1 whose only positive divisors are 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23.
Primes are the multiplicative building blocks of the integers. The Fundamental Theorem of Arithmetic says every integer greater than 1 has a unique factorization into primes (up to ordering). There are infinitely many primes — Euclid's proof from around 300 BCE remains one of the most elegant arguments in mathematics. The distribution of primes thins out logarithmically: by the Prime Number Theorem, the number of primes below \(n\) is approximately \(n / \ln n\).
Primes are central to modern cryptography (RSA, elliptic-curve), and the search for very large primes (particularly Mersenne primes) has driven distributed computing projects like GIMPS for decades.