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text: 'A [-45h] $n > 1$ is *prime* if it has no [divisor_number_theory divisors] other than itself and $1$.\nEquivalently, it has the property that if $n \\mid ab$ %%note:That is, $n$ divides the product $ab$%% then $n \\mid a$ or $n \\mid b$.\nConventionally, $1$ is considered to be neither prime nor [composite_number composite] (i.e. non-prime).\n\n# Examples\n\n- The number $2$ is prime, because its divisors are $1$ and $2$; therefore it has no divisors other than itself and $1$.\n- The number $3$ is also prime, as are $5, 7, 11, 13, \\dots$.\n- The number $4$ is not prime; neither are $6, 8, 9, 10, 12, \\dots$.\n\n# Properties\n\n- There are infinitely many primes. ([54r Proof.])\n- Every natural number may be written as a product of primes; moreover, this can only be done in one way (if we count "the same product but with the order swapped" as being the same: for example, $2 \\times 3 = 3 \\times 2$ is just one way of writing $6$). ([fundamental_theorem_of_arithmetic Proof.])\n\n# How to find primes\n\nIf we want to create a list of all the primes below a given number, or the first $n$ primes for some fixed $n$, then an efficient way to do it is the [sieve_of_eratosthenes Sieve of Eratosthenes].\n(There are other sieves available, but Eratosthenes is the simplest.)\n\nThere are many [primality_testing tests] for primality and for compositeness.\n\n# More general concept\n\nThis definition of "prime" is, in a more general [3gq ring-theoretic] setting, known instead as the property of [5m1 irreducibility].\nConfusingly, there is a slightly different notion in this ring-theoretic setting, which goes by the name of "prime"; this notion has [prime_element_ring_theory a separate page on Arbital].\nIn the ring of integers, the two ideas of "prime" and "irreducible" actually coincide, but that is because the integers form a ring with several very convenient properties: in particular, being a [euclidean_domain Euclidean domain], they are a [-principal_ideal_domain] (PID), and [pid_implies_ufd PIDs have unique factorisation].\n\n[todo: add requisite for divisor_number_theory]',
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