Membership in co-NP is also straightforward: one can just list the prime factors of m, which the verifier can confirm to be valid by multiplication and the AKS primality test.

This is because both YES and NO answers can be verified in polynomial time given the prime factors ( we can verify their primality using the AKS primality test, and that their product is N by multiplication ).

") appears to be much easier than the problem of actually finding the factors of N. Specifically, the former can be solved in polynomial time ( in the number n of digits of N ) with the AKS primality test.

# Test that F ≠ N, that F divides N ( time complexity O ( log N )), and that F is prime ( polynomial time ; see AKS primality test ).

( A polynomial-time primality test which does not require GRH, the AKS primality test, was published in 2002.

The AKS primality test, runs in Õ (( log n )< sup > 12 </ sup >) ( improved to Õ (( log n )< sup > 7. 5 </ sup >) in the published revision of their paper ), which can be further reduced to Õ (( log n )< sup > 6 </ sup >) if the Sophie Germain conjecture is true.

The existence of the AKS primality test finally settled this long-standing question and placed PRIMES in P. However, PRIMES is not known to be P-complete, and it is not known whether it lies in classes lying inside P such as NC or L.

For theoretical purposes, it was superseded by the AKS primality test, which does not rely on unproven assumptions.

The AKS primality test ( also known as Agrawal – Kayal – Saxena primality test and cyclotomic AKS test ) is a deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002, in a paper titled " PRIMES is in P ".

* The AKS algorithm can be used to verify the primality of any general number given.

The AKS primality test is based upon the following theorem: An integer n (≥ 2 ) is prime if and only if the polynomial congruence relation

# REDIRECT AKS primality test

# REDIRECT AKS primality test

# REDIRECT AKS primality test

# REDIRECT AKS primality test

# REDIRECT AKS primality test

** Manindra Agrawal, Neeraj Kayal and Nitin Saxena, for the AKS primality test.

AKS primality testing is such an algorithm with polynomial running time.

The first conjecture ( Agrawal ’ s conjecture ) was the basis for the formulation of the first deterministic prime test algorithm in polynomial time ( AKS algorithm ).

Also in 1972, Synthi AKS was released, and its digital sequencer with a touch-sensitive flat keyboard, KS sequencer, and its mechanical keyboard version, DKS, were also released.

The AKS also has a sequencer built into the keyboard in the lid.

The nationalistic ideology of the AKS also stemmed from the common European discussion of national rights based on the 14 points of President Wilson.

AKS also organized aid to Finnic minorities in Soviet Russia and refugees from there and promoted cultural efforts to help the Finnish-speaking minorities of northern Sweden and Norway.

AKS also maintained close ties with a militant secret society called Vihan Veljet.

Another online resource in Korean about Kim Jeong-hui is the page about him in the EncyKorea ( also a resource administered by the AKS ).

* AK Steel Holding Company, a US-based S & P 500 ( NYSE: AKS ) Steel Manufacturer

In response to some of these variants, and to other feedback, the paper " PRIMES is in P " was updated with a new formulation of the AKS algorithm and of its proof of correctness.

* The AKS " PRIMES in P " Algorithm Resource

The Under 16 team defeated ISD while the Under 18 team defeated Aga Khan School. The Girls team lost to AKS in the finals and became the runners-up.

AKS is the first primality-proving algorithm to be simultaneously general, polynomial, deterministic, and unconditional.

In 2005, Carl Pomerance and H. W. Lenstra, Jr. demonstrated a variant of AKS that runs in Õ ( log < sup > 6 </ sup >( n )) operations, where n is the number to be tested – a marked improvement over the initial Õ ( log < sup > 12 </ sup >( n )) bound in the original algorithm.

* JAVA implementation of the AKS Primality Test algorithm.

Historically notable is the former club AKS Chorzów.

During the making of their third album, they purchased their first commercial synthesisers for the studio, the Minimoog and EMS Synthi AKS.

* The correctness of AKS is not conditional on any subsidiary unproven hypothesis.

The proof of correctness for AKS consists of showing that there exists a suitably small r and suitably small set of integers A such that, if the congruence holds for all such a in A, then n must be prime.

Models ' early style was a spiky, distinctive blend of New Wave, glam rock, dub and pop: which included Kelly's strangled singing voice, Duffield's virtuoso synthesiser performances ( he used the EMS Synthi AKS, as used by Pere Ubu on their Dub Housing ), and the band's cryptic, slightly gruesome, lyrics ( e. g. " Hans Stand: A War Record " from Alphabravocharliedeltaechofoxtrotgolf ), which were mostly written or co-written by Kelly.

Carmichael numbers are important because they pass the Fermat primality test but are not actually prime.

Since Carmichael numbers exist, this primality test cannot be relied upon to prove the primality of a number, although it can still be used to prove a number is composite.

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

To perform its testing, the project relies primarily on Édouard Lucas and Derrick Henry Lehmer's primality test, an algorithm that is both specialized to testing Mersenne primes and particularly efficient on binary computer architectures.

In addition, there are a number of probabilistic algorithms that can test primality very quickly in practice if one is willing to accept the vanishingly small possibility of error.

The Lucas – Lehmer primality test ( LLT ) is an efficient primality test that greatly aids this task.

The best method presently known for testing the primality of Mersenne numbers is the Lucas – Lehmer primality test.

Algorithms that are much more efficient than trial division have been devised to test the primality of large numbers.

Prime integers can be efficiently found using a primality test.

In other words, a composite integer is a Fermat pseudoprime to base a if it successfully passes Fermat primality test for the base a.

Fermat primality test for the base 2.

Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory.

This theorem forms the basis for the Lucas – Lehmer test, an important primality test.