Alkulukut

Alkulukut, the Finnish term for prime numbers, are the natural numbers greater than 1 that have no positive divisors other than 1 and themselves. The smallest alkulukut are 2, 3, 5, 7, and 11. The number 1 is not an alkulukku by definition. By contrast, numbers greater than 1 that are not alkulukut are called composite.

Fundamental to number theory is the fact that every integer greater than 1 is either an alkulukku or a product of alkulukut in a way that is unique up to the order of the factors. This fundamental theorem of arithmetic underpins many mathematical results and algorithms. There are infinitely many alkulukut; Euclid gave a classic proof by contradiction showing that no finite list of alkulukut can exhaust them.

Primality testing and generation of alkulukut are central computational tasks. Methods include trial division by primes up to the square root of the number and sieving techniques such as the Sieve of Eratosthenes for finding all alkulukut up to a limit. More advanced tests, like Miller–Rabin, are used for large numbers in practical applications, with deterministic variants available for certain ranges. Alkulukut are also fundamental in cryptography, particularly in public-key schemes such as RSA, which rely on the properties of large alkulukut and the difficulty of factoring their products.

Distribution and patterns of alkulukut are active areas of study. The prime number theorem shows that the number of alkulukut less than x is asymptotically x divided by the natural logarithm of x, informing expectations about their density. Research continues on topics such as prime gaps and conjectures about twin primes.