Primes

Primes are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. In other words, a prime can be divided evenly only by 1 and by the number. The first primes are 2, 3, 5, 7, 11, and 13. The number 1 is not prime, and every even number greater than 2 is composite because it has a divisor of 2.

One of the central ideas in number theory is that primes are the building blocks of the integers. Any integer greater than 1 can be factored uniquely into prime factors, up to the order of the factors. This is the Fundamental Theorem of Arithmetic.

There are infinitely many primes; this was proven by Euclid. Primes become less frequent as numbers grow larger, though they never cease entirely. The Prime Number Theorem gives a precise description of their distribution: the number of primes less than or equal to x is asymptotically x / log x. The gaps between consecutive primes vary and can be arbitrarily large.

Notable topics include special classes of primes, such as twin primes (pairs of primes differing by 2), for which it is conjectured that infinitely many exist. Primes of the form 2^p − 1, known as Mersenne primes, are studied for their connection to even perfect numbers. In modern applications, primes play a key role in cryptography, particularly in algorithms that rely on the difficulty of factorizing large numbers.