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A commutative ring is an algebraic structure where addition, subtraction and multiplication are defined.
The integers are a ring, and the prime numbers in the integers have been generalized to rings in two different ways, prime elements and irreducible elements.
An element is irreducible if it is neither a unit nor the product of two other non-unit elements. In the ring of integers, the prime and irreducible elements form the same set,.
In an arbitrary ring, all prime elements are irreducible. The converse does not hold in general, but does hold for unique factorization domains.
The fundamental theorem of arithmetic continues to hold by definition in unique factorization domains. Its prime elements are known as Gaussian primes.
Rational primes the prime elements in the integers congruent to 3 mod 4 are Gaussian primes, but rational primes congruent to 1 mod 4 are not.
Not every ring is a unique factorization domain. In order to extend unique factorization to a larger class of rings, the notion of a number can be replaced with that of an ideal , a subset of the elements of a ring that contains all sums of pairs of its elements, and all products of its elements with ring elements.
Prime ideals , which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra , algebraic number theory and algebraic geometry.
The prime ideals of the ring of integers are the ideals 0 , 2 , 3 , 5 , 7 , 11 , … The fundamental theorem of arithmetic generalizes to the Lasker—Noether theorem , which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals , which are the appropriate generalizations of prime powers.
The spectrum of a ring is a geometric space whose points are the prime ideals of the ring. For example, factorization or ramification of prime ideals when lifted to an extension field , a basic problem of algebraic number theory, bears some resemblance with ramification in geometry.
These concepts can even assist with in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity , a statement that concerns the existence of square roots modulo integer prime numbers.
By Lagrange's theorem , any group of prime order is a cyclic group , and by the Burnside theorem any group whose order is divisible by only two primes is solvable.
For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of mathematics [b] with the exception of use of prime numbered gear teeth to distribute wear evenly.
Hardy prided themselves on doing work that had absolutely no military significance. This vision of the purity of number theory was shattered in the s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms.
The most basic primality testing routine, trial division, is too slow to be useful for large numbers. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for numbers of special types.
Most primality tests only tell whether their argument is prime or not. Routines that also provide a prime factor of composite arguments or all of its prime factors are called factorization algorithms.
Prime numbers are also used in computing for checksums , hash tables , and pseudorandom number generators. Another optimization is to check only primes as factors in this range.
Each division produces a nonzero remainder, so 37 is indeed prime. Although this method is simple to describe, it is impractical for testing the primality of large integers, because the number of tests that it performs grows exponentially as a function of the number of digits of these integers.
Before computers, mathematical tables listing all of the primes or prime factorizations up to a given limit were commonly printed.
Because this decreases exponentially with the number of tests, it provides high confidence although not certainty that a number that passes the repeated test is prime.
On the other hand, if the test ever fails, then the number is certainly composite. In contrast, some other algorithms guarantee that their answer will always be correct: For instance, this is true of trial division.
The algorithms with guaranteed-correct output include both deterministic non-random algorithms, such as the AKS primality test ,  and randomized Las Vegas algorithms where the random choices made by the algorithm do not affect its final answer, such as some variations of elliptic curve primality proving.
The AKS primality test has mathematically proven time complexity, but is slower than elliptic curve primality proving in practice. The following table lists some of these tests.
In addition to the aforementioned tests that apply to any natural number, some numbers of a special form can be tested for primality more quickly. For example, the Lucas—Lehmer primality test can determine whether a Mersenne number one less than a power of two is prime, deterministically, in the same time as a single iteration of the Miller—Rabin test.
The following table gives the largest known primes of various types. Some of these primes have been found using distributed computing. It is significantly more difficult than primality testing,  and although many factorization algorithms are known, they are slower than the fastest primality testing methods.
As with primality testing, there are also factorization algorithms that require their input to have a special form, including the special number field sieve.
Shor's algorithm can factor any integer in a polynomial number of steps on a quantum computer.
Several public-key cryptography algorithms, such as RSA and the Diffie—Hellman key exchange , are based on large prime numbers bit primes are common.
Prime numbers are frequently used for hash tables. For instance the original method of Carter and Wegman for universal hashing was based on computing hash functions by choosing random linear functions modulo large prime numbers.
Some checksum methods are based on the mathematics of prime numbers. For instance the checksums used in International Standard Book Numbers are defined by taking the rest of the number modulo 11, a prime number.
Because 11 is prime this method can detect both single-digit errors and transpositions of adjacent digits. Prime numbers are of central importance to number theory but also have many applications to other areas within mathematics, including abstract algebra and elementary geometry.
For example, it is possible to place prime numbers of points in a two-dimensional grid so that no three are in a line , or so that every triangle formed by three of the points has large area.
The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics.
Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field of a given field is its smallest subfield that contains both 0 and 1.
It is either the field of rational numbers or a finite field with a prime number of elements, whence the name. For example, in knot theory , a prime knot is a knot that is indecomposable in the sense that it cannot be written as the connected sum of two nontrivial knots.
Any knot can be uniquely expressed as a connected sum of prime knots. Beyond mathematics and computing, prime numbers have potential connections to quantum mechanics , and have been used metaphorically in the arts and literature.
They have also been used in evolutionary biology to explain the life cycles of cicadas. Fermat primes are primes of the form.
Beginning with the work of Hugh Montgomery and Freeman Dyson in the s, mathematicians and physicists have speculated that the zeros of the Riemann zeta function are connected to the energy levels of quantum systems.
The evolutionary strategy used by cicadas of the genus Magicicada makes use of prime numbers. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most.
Biologists theorize that these prime-numbered breeding cycle lengths have evolved in order to prevent predators from synchronizing with these cycles.
Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena".
According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".
In his science fiction novel Contact , scientist Carl Sagan suggested that prime factorization could be used as a means of establishing two-dimensional image planes in communications with aliens, an idea that he had first developed informally with American astronomer Frank Drake in From Wikipedia, the free encyclopedia.
Redirected from Prime numbers. For other uses, see Prime disambiguation. List of prime numbers.
Fundamental theorem of arithmetic. Conjectures about prime numbers. Prime number theorem and Prime-counting function.
Prime element and Irreducible element. The sieve of Eratosthenes starts with all numbers unmarked gray. It repeatedly finds the first unmarked number, marks it as prime dark colors and marks its square and all later multiples as composite lighter colors.
After marking the multiples of 2 red , 3 green , 5 blue , and 7 yellow , all primes up to the square root of the table size have been processed, and all remaining unmarked numbers 11, 13, etc.
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