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Title: AKS primality test - Wikipedia

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(Top) https://en.wikipedia.org/wiki/AKS_primality_test
1 Importance https://en.wikipedia.org/wiki/AKS_primality_test#Importance
2 Concepts https://en.wikipedia.org/wiki/AKS_primality_test#Concepts
3 History and running time https://en.wikipedia.org/wiki/AKS_primality_test#History_and_running_time
4 The algorithm https://en.wikipedia.org/wiki/AKS_primality_test#The_algorithm
4.1 Proof of validity outline https://en.wikipedia.org/wiki/AKS_primality_test#Proof_of_validity_outline
4.2 Example 1: n = 31 is prime https://en.wikipedia.org/wiki/AKS_primality_test#Example_1:_n_=_31_is_prime
5 References https://en.wikipedia.org/wiki/AKS_primality_test#References
6 Further reading https://en.wikipedia.org/wiki/AKS_primality_test#Further_reading
7 External links https://en.wikipedia.org/wiki/AKS_primality_test#External_links
العربيةhttps://ar.wikipedia.org/wiki/%D8%A7%D8%AE%D8%AA%D8%A8%D8%A7%D8%B1_%D8%A3.%D9%83.%D8%A3%D8%B3_%D9%84%D8%A3%D9%88%D9%84%D9%8A%D8%A9_%D8%B9%D8%AF%D8%AF_%D9%85%D8%A7
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日本語https://ja.wikipedia.org/wiki/AKS%E7%B4%A0%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95
Portuguêshttps://pt.wikipedia.org/wiki/Teste_de_primalidade_AKS
Deutschhttps://de.wikipedia.org/wiki/AKS-Primzahltest
Françaishttps://fr.wikipedia.org/wiki/Test_de_primalit%C3%A9_AKS
Españolhttps://es.wikipedia.org/wiki/Test_de_primalidad_AKS
മലയാളംhttps://ml.wikipedia.org/wiki/%E0%B4%8E.%E0%B4%95%E0%B5%86.%E0%B4%8E%E0%B4%B8%E0%B5%8D._%E0%B4%85%E0%B4%AD%E0%B4%BE%E0%B4%9C%E0%B5%8D%E0%B4%AF%E0%B4%A4%E0%B4%BE%E0%B4%AA%E0%B4%B0%E0%B4%BF%E0%B4%B6%E0%B5%8B%E0%B4%A7%E0%B4%A8
Polskihttps://pl.wikipedia.org/wiki/Test_pierwszo%C5%9Bci_AKS
Nederlandshttps://nl.wikipedia.org/wiki/AKS-test
한국어https://ko.wikipedia.org/wiki/AKS_%EC%86%8C%EC%88%98%ED%8C%90%EB%B3%84%EB%B2%95
Italianohttps://it.wikipedia.org/wiki/Algoritmo_AKS
עבריתhttps://he.wikipedia.org/wiki/%D7%9E%D7%91%D7%97%D7%9F_AKS_%D7%9C%D7%A8%D7%90%D7%A9%D7%95%D7%A0%D7%99%D7%95%D7%AA
فارسیhttps://fa.wikipedia.org/wiki/%D8%A2%D8%B2%D9%85%D9%88%D9%86_%D8%A7%D9%88%D9%84_%D8%A8%D9%88%D8%AF%D9%86_%D8%A7%DB%8C%E2%80%8C%DA%A9%DB%8C%E2%80%8C%D8%A7%D8%B3
Українськаhttps://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%81%D1%82_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%82%D0%B8_AKS
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Esperantohttps://eo.wikipedia.org/wiki/Primeco-testo_AKS
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Wikidata itemhttps://www.wikidata.org/wiki/Special:EntityPage/Q294284
deterministichttps://en.wikipedia.org/wiki/Deterministic_algorithm
primality-provinghttps://en.wikipedia.org/wiki/Primality_test
algorithmhttps://en.wikipedia.org/wiki/Algorithm
Manindra Agrawalhttps://en.wikipedia.org/wiki/Manindra_Agrawal
Neeraj Kayalhttps://en.wikipedia.org/wiki/Neeraj_Kayal
Nitin Saxenahttps://en.wikipedia.org/wiki/Nitin_Saxena
Indian Institute of Technology Kanpurhttps://en.wikipedia.org/wiki/Indian_Institute_of_Technology_Kanpur
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
polynomial timehttps://en.wikipedia.org/wiki/Polynomial_time
primehttps://en.wikipedia.org/wiki/Prime_number
compositehttps://en.wikipedia.org/wiki/Composite_number
mathematical conjectureshttps://en.wikipedia.org/wiki/Mathematical_conjecture
generalized Riemann hypothesishttps://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
analysishttps://en.wikipedia.org/wiki/Analysis_(mathematics)
[2]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-2
Gödel Prizehttps://en.wikipedia.org/wiki/G%C3%B6del_Prize
Fulkerson Prizehttps://en.wikipedia.org/wiki/Fulkerson_Prize
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=1
Lucas–Lehmer testhttps://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test
Mersenne numbershttps://en.wikipedia.org/wiki/Mersenne_number
Pépin's testhttps://en.wikipedia.org/wiki/P%C3%A9pin%27s_test
Fermat numbershttps://en.wikipedia.org/wiki/Fermat_number
polynomialhttps://en.wikipedia.org/wiki/Polynomial_time#Polynomial_time
ECPPhttps://en.wikipedia.org/wiki/Elliptic_curve_primality_proving
APRhttps://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test
deterministicallyhttps://en.wikipedia.org/wiki/Deterministic_algorithm
Miller–Rabinhttps://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
Baillie–PSWhttps://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test
hypothesishttps://en.wikipedia.org/wiki/Hypothesis
Miller–Rabin testhttps://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants
generalized Riemann hypothesishttps://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis
galactic algorithmhttps://en.wikipedia.org/wiki/Galactic_algorithm
Baillie–PSW testhttps://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test
primality certificatehttps://en.wikipedia.org/wiki/Primality_certificate
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=2
coprimehttps://en.wikipedia.org/wiki/Coprime
polynomialhttps://en.wikipedia.org/wiki/Polynomial_ring
congruence relationhttps://en.wikipedia.org/wiki/Congruence_relation#Modular_arithmetic
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
indeterminatehttps://en.wikipedia.org/wiki/Polynomial_ring#Definition_(univariate_case)
Fermat's little theoremhttps://en.wikipedia.org/wiki/Fermat%27s_little_theorem
binomial theoremhttps://en.wikipedia.org/wiki/Binomial_theorem
binomial coefficienthttps://en.wikipedia.org/wiki/Binomial_coefficient
1https://en.wikipedia.org/wiki/AKS_primality_test#math_1
exponential timehttps://en.wikipedia.org/wiki/Exponential_time
brute forcehttps://en.wikipedia.org/wiki/Brute_force_method
polynomial ringhttps://en.wikipedia.org/wiki/Polynomial_ring
computational complexityhttps://en.wikipedia.org/wiki/Computational_complexity_theory
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
3https://en.wikipedia.org/wiki/AKS_primality_test#math_3
1https://en.wikipedia.org/wiki/AKS_primality_test#math_1
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=3
Õhttps://en.wikipedia.org/wiki/Big_O_notation#Extensions_to_the_Bachmann–Landau_notations
big O notationhttps://en.wikipedia.org/wiki/Big_O_notation
Sophie Germain primeshttps://en.wikipedia.org/wiki/Sophie_Germain_prime
Annals of Mathematicshttps://en.wikipedia.org/wiki/Annals_of_Mathematics
cyclotomic polynomialshttps://en.wikipedia.org/wiki/Cyclotomic_polynomial
finite fieldshttps://en.wikipedia.org/wiki/Finite_fields
sieve theoryhttps://en.wikipedia.org/wiki/Sieve_theory
Pomerancehttps://en.wikipedia.org/wiki/Carl_Pomerance
Lenstrahttps://en.wikipedia.org/wiki/Hendrik_Lenstra
[3]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-lenstra_pomerance_2005-3
[4]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-lenstra_pomerance_2011-4
Agrawal's conjecturehttps://en.wikipedia.org/wiki/Agrawal%27s_conjecture
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=4
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
perfect powerhttps://en.wikipedia.org/wiki/Perfect_power
ordhttps://en.wikipedia.org/wiki/Multiplicative_order
ordhttps://en.wikipedia.org/wiki/Multiplicative_order
multiplicative orderhttps://en.wikipedia.org/wiki/Multiplicative_order
modulohttps://en.wikipedia.org/wiki/Modular_arithmetic
binary logarithmhttps://en.wikipedia.org/wiki/Binary_logarithm
Euler's totient functionhttps://en.wikipedia.org/wiki/Euler%27s_totient_function
gcdhttps://en.wikipedia.org/wiki/Greatest_common_divisor
monomialshttps://en.wikipedia.org/wiki/Monomial
coefficientshttps://en.wikipedia.org/wiki/Coefficient
[5]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-bernstein03-5
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multiplicative grouphttps://en.wikipedia.org/wiki/Multiplicative_group
contradictionhttps://en.wikipedia.org/wiki/Proof_by_contradiction
[1]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-AKS-1
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=6
gcdhttps://en.wikipedia.org/wiki/Greatest_common_divisor
PolynomialRemainderhttps://en.wikipedia.org/wiki/Polynomial_remainder
Ahttps://en.wikipedia.org/wiki/AKS_primality_test#math_A
Bhttps://en.wikipedia.org/wiki/AKS_primality_test#math_B
[6]https://en.wikipedia.org/wiki/AKS_primality_test#cite_note-6
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"PRIMES is in P"http://www.cse.iitk.ac.in/users/manindra/algebra/primality_v6.pdf
Annals of Mathematicshttps://en.wikipedia.org/wiki/Annals_of_Mathematics
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.4007/annals.2004.160.781https://doi.org/10.4007%2Fannals.2004.160.781
JSTORhttps://en.wikipedia.org/wiki/JSTOR_(identifier)
3597229https://www.jstor.org/stable/3597229
^https://en.wikipedia.org/wiki/AKS_primality_test#cite_ref-2
"It is easy to determine whether a given integer is prime"https://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/home.html
doihttps://en.wikipedia.org/wiki/Doi_(identifier)
10.1090/S0273-0979-04-01037-7https://doi.org/10.1090%2FS0273-0979-04-01037-7
^https://en.wikipedia.org/wiki/AKS_primality_test#cite_ref-lenstra_pomerance_2005_3-0
Primality testing with Gaussian periodshttp://www.math.dartmouth.edu/~carlp/PDF/complexity12.pdf
^https://en.wikipedia.org/wiki/AKS_primality_test#cite_ref-lenstra_pomerance_2011_4-0
Primality testing with Gaussian periodshttp://www.math.dartmouth.edu/~carlp/aks041411.pdf
Archivedhttps://web.archive.org/web/20120225052810/http://www.math.dartmouth.edu/~carlp/aks041411.pdf
Wayback Machinehttps://en.wikipedia.org/wiki/Wayback_Machine
^https://en.wikipedia.org/wiki/AKS_primality_test#cite_ref-bernstein03_5-0
Proving Primality After Agrawal-Kayal-Saxenahttps://cr.yp.to/papers/aks.pdf
^https://en.wikipedia.org/wiki/AKS_primality_test#cite_ref-6
AKS Talkhttps://en.wikipedia.org/wiki/Talk:AKS_primality_test#Worked_Example
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=8
Springer-Verlaghttps://en.wikipedia.org/wiki/Springer-Verlag
ISBNhttps://en.wikipedia.org/wiki/ISBN_(identifier)
3-540-40344-2https://en.wikipedia.org/wiki/Special:BookSources/3-540-40344-2
Zblhttps://en.wikipedia.org/wiki/Zbl_(identifier)
1058.11070https://zbmath.org/?format=complete&q=an:1058.11070
edithttps://en.wikipedia.org/w/index.php?title=AKS_primality_test&action=edit§ion=9
Weisstein, Eric W.https://en.wikipedia.org/wiki/Eric_W._Weisstein
"AKS Primality Test"https://mathworld.wolfram.com/AKSPrimalityTest.html
MathWorldhttps://en.wikipedia.org/wiki/MathWorld
R. Crandall, Apple ACG, and J. Papadopoulos (March 18, 2003): On the implementation of AKS-class primality testshttps://web.archive.org/web/20140219064936/http://www.dm.unito.it/~cerruti/ac/aks-crandall.pdf
Article by Bornemann, containing photos and information about the three Indian scientistshttps://www.ams.org/notices/200305/fea-bornemann.pdf
Andrew Granville: It is easy to determine whether a given integer is primehttps://www.ams.org/bull/2005-42-01/S0273-0979-04-01037-7/home.html
The Prime Facts: From Euclid to AKShttp://www.scottaaronson.com/writings/prime.pdf
Scott Aaronsonhttps://en.wikipedia.org/wiki/Scott_Aaronson
The PRIMES is in P little FAQhttp://www.instantlogic.net/publications/PRIMES%20is%20in%20P%20little%20FAQ.htm
2006 Gödel Prize Citationhttps://web.archive.org/web/20150327071905/http://www.sigact.org/Prizes/Godel/2006.html
2006 Fulkerson Prize Citationhttps://www.ams.org/notices/200611/comm-fulkerson.pdf
The AKS "PRIMES in P" Algorithm Resourcehttp://fatphil.org/maths/AKS
vhttps://en.wikipedia.org/wiki/Template:Number-theoretic_algorithms
thttps://en.wikipedia.org/wiki/Template_talk:Number-theoretic_algorithms
ehttps://en.wikipedia.org/wiki/Special:EditPage/Template:Number-theoretic_algorithms
Number-theoretichttps://en.wikipedia.org/wiki/Number_theory
algorithmshttps://en.wikipedia.org/wiki/Algorithm
Primality testshttps://en.wikipedia.org/wiki/Primality_test
APRhttps://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%80%93Rumely_primality_test
Baillie–PSWhttps://en.wikipedia.org/wiki/Baillie%E2%80%93PSW_primality_test
Elliptic curvehttps://en.wikipedia.org/wiki/Elliptic_curve_primality
Pocklingtonhttps://en.wikipedia.org/wiki/Pocklington_primality_test
Fermathttps://en.wikipedia.org/wiki/Fermat_primality_test
Lucashttps://en.wikipedia.org/wiki/Lucas_primality_test
Lucas–Lehmerhttps://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer_primality_test
Lucas–Lehmer–Rieselhttps://en.wikipedia.org/wiki/Lucas%E2%80%93Lehmer%E2%80%93Riesel_test
Proth's theoremhttps://en.wikipedia.org/wiki/Proth%27s_theorem
Pépin'shttps://en.wikipedia.org/wiki/P%C3%A9pin%27s_test
Quadratic Frobeniushttps://en.wikipedia.org/wiki/Quadratic_Frobenius_test
Solovay–Strassenhttps://en.wikipedia.org/wiki/Solovay%E2%80%93Strassen_primality_test
Miller–Rabinhttps://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test
Prime-generatinghttps://en.wikipedia.org/wiki/Generating_primes
Sieve of Atkinhttps://en.wikipedia.org/wiki/Sieve_of_Atkin
Sieve of Eratostheneshttps://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
Sieve of Pritchardhttps://en.wikipedia.org/wiki/Sieve_of_Pritchard
Sieve of Sundaramhttps://en.wikipedia.org/wiki/Sieve_of_Sundaram
Wheel factorizationhttps://en.wikipedia.org/wiki/Wheel_factorization
Integer factorizationhttps://en.wikipedia.org/wiki/Integer_factorization
Continued fraction (CFRAC)https://en.wikipedia.org/wiki/Continued_fraction_factorization
Dixon'shttps://en.wikipedia.org/wiki/Dixon%27s_factorization_method
Lenstra elliptic curve (ECM)https://en.wikipedia.org/wiki/Lenstra_elliptic-curve_factorization
Euler'shttps://en.wikipedia.org/wiki/Euler%27s_factorization_method
Pollard's rhohttps://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm
p − 1https://en.wikipedia.org/wiki/Pollard%27s_p_%E2%88%92_1_algorithm
p + 1https://en.wikipedia.org/wiki/Williams%27s_p_%2B_1_algorithm
Quadratic sieve (QS)https://en.wikipedia.org/wiki/Quadratic_sieve
General number field sieve (GNFS)https://en.wikipedia.org/wiki/General_number_field_sieve
Special number field sieve (SNFS)https://en.wikipedia.org/wiki/Special_number_field_sieve
Rational sievehttps://en.wikipedia.org/wiki/Rational_sieve
Fermat'shttps://en.wikipedia.org/wiki/Fermat%27s_factorization_method
Shanks's square formshttps://en.wikipedia.org/wiki/Shanks%27s_square_forms_factorization
Trial divisionhttps://en.wikipedia.org/wiki/Trial_division
Shor'shttps://en.wikipedia.org/wiki/Shor%27s_algorithm
Multiplicationhttps://en.wikipedia.org/wiki/Multiplication_algorithm
Ancient Egyptianhttps://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication
Longhttps://en.wikipedia.org/wiki/Long_multiplication
Karatsubahttps://en.wikipedia.org/wiki/Karatsuba_algorithm
Toom–Cookhttps://en.wikipedia.org/wiki/Toom%E2%80%93Cook_multiplication
Schönhage–Strassenhttps://en.wikipedia.org/wiki/Sch%C3%B6nhage%E2%80%93Strassen_algorithm
Fürer'shttps://en.wikipedia.org/wiki/F%C3%BCrer%27s_algorithm
Euclideanhttps://en.wikipedia.org/wiki/Euclidean_division
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