Here's a comprehensive summary of the Wikipedia article on prime numbers:

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers; otherwise, it's called a composite number. Primes are central to number theory because, according to the fundamental theorem of arithmetic, every natural number greater than 1 is either a prime itself or can be uniquely factored as a product of primes.

While methods like trial division can check primality, faster algorithms exist for larger numbers, and as of October 2024, the largest known prime is a Mersenne prime with over 41 million decimal digits. Euclid demonstrated around 300 BC that there are infinitely many primes, and their statistical distribution within natural numbers is described by the Prime Number Theorem, proven in the late 19th century.

Despite extensive research, several historical questions remain unsolved, including Goldbach's conjecture (that every even integer greater than 2 is the sum of two primes) and the twin prime conjecture (that there are infinitely many pairs of primes differing by two). Crucially, prime numbers are vital in modern information technology, especially in public-key cryptography, which relies on the computational difficulty of factoring large numbers into their prime components.