{"history":[],"article":{"id":"d6acd3b1-34d0-46b7-9387-e96a38a874d2","title":"AKS primality test","slug":"aks-primality-test","content":":::summary\n:::\n\n# \nThe **AKS primality test** (also known as **Agrawal–Kayal–Saxena primality test** and **cyclotomic AKS test**) is a [deterministic](https://bliptext.com/articles/deterministic-[algorithm](https://bliptext.com/articles/algorithm)) [primality-proving](https://bliptext.com/articles/primality-test) [algorithm](https://bliptext.com/articles/algorithm) created and published by [Manindra Agrawal](https://bliptext.com/articles/manindra-agrawal), [Neeraj Kayal](https://bliptext.com/articles/neeraj-kayal), and [Nitin Saxena](https://bliptext.com/articles/nitin-saxena), computer scientists at the [Indian Institute of Technology Kanpur](https://bliptext.com/articles/indian-institute-of-technology-kanpur), on August 6, 2002, in an article titled \"PRIMES is in P\".\nThe algorithm was the first one which is able to determine in [polynomial time](https://bliptext.com/articles/polynomial-time), whether a given number is [prime](https://bliptext.com/articles/prime-number) or [composite](https://bliptext.com/articles/composite-number) without relying on [mathematical conjectures](https://bliptext.com/articles/mathematical-conjecture) such as the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis).\nThe proof is also notable for not relying on the field of [analysis](https://bliptext.com/articles/analysis-mathematics).\nIn 2006 the authors received both the [Gödel Prize](https://bliptext.com/articles/g-del-prize) and [Fulkerson Prize](https://bliptext.com/articles/fulkerson-prize) for their work.\n\n# Importance\nAKS is the first primality-proving algorithm to be simultaneously general, polynomial-time, deterministic, and unconditionally correct.\nPrevious algorithms had been developed for centuries and achieved three of these properties at most, but not all four.\nWhile the algorithm is of immense theoretical importance, it is not used in practice, rendering it a [galactic algorithm](https://bliptext.com/articles/galactic-algorithm).\nFor 64-bit inputs, the [Baillie–PSW test](https://bliptext.com/articles/baillie-psw-primality-test) is deterministic and runs many orders of magnitude faster.\nFor larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is far superior to AKS.\nAdditionally, ECPP can output a [primality certificate](https://bliptext.com/articles/primality-certificate) that allows independent and rapid verification of the results, which is not possible with the AKS algorithm.\n\n# Concepts\nThe AKS primality test is based upon the following theorem: Given an integer n\\ge 2 and integer a [coprime](https://bliptext.com/articles/coprime) to n, n is prime if and only if the [polynomial](https://bliptext.com/articles/polynomial-ring) [congruence relation](https://bliptext.com/articles/congruence-relation)\nholds within the polynomial ring.\nNote that X denotes the [indeterminate](https://bliptext.com/articles/polynomial-ring) which generates this polynomial ring.\nThis theorem is a generalization to polynomials of [Fermat's little theorem](https://bliptext.com/articles/fermat-s-little-theorem).\nIn one direction it can easily be proven using the [binomial theorem](https://bliptext.com/articles/binomial-theorem) together with the following property of the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient):\nWhile the relation constitutes a primality test in itself, verifying it takes [exponential time](https://bliptext.com/articles/exponential-time): the [brute force](https://bliptext.com/articles/brute-force-method) approach would require the expansion of the (X + a)^n polynomial and a reduction \\pmod{n} of the resulting n + 1 coefficients.\nThe congruence is an equality in the [polynomial ring](https://bliptext.com/articles/polynomial-ring).\nEvaluating in a quotient ring of creates an upper bound for the degree of the polynomials involved.\nThe AKS evaluates the equality in, making the [computational complexity](https://bliptext.com/articles/computational-complexity-theory) dependent on the size of r. For clarity, this is expressed as the congruence\nwhich is the same as:\nfor some polynomials f and g.\nNote that all primes satisfy this relation (choosing g=0 in gives, which holds for n prime).\nThis congruence can be checked in polynomial time when r is polynomial to the digits of n. The AKS algorithm evaluates this congruence for a large set of a values, whose size is polynomial to the digits of n. The proof of validity of the AKS algorithm shows that one can find an r and a set of a values with the above properties such that if the congruences hold then n is a power of a prime.\n\n# History and running time\nIn the first version of the above-cited paper, the authors proved the asymptotic time complexity of the algorithm to be (using [Õ](https://bliptext.com/articles/big-o-notation) from [big O notation](https://bliptext.com/articles/big-o-notation))—the twelfth power of the number of digits in n times a factor that is polylogarithmic in the number of digits.\nHowever, this upper bound was rather loose; a widely-held conjecture about the distribution of the [Sophie Germain primes](https://bliptext.com/articles/sophie-germain-prime) would, if true, immediately cut the worst case down to.\nIn the months following the discovery, new variants appeared (Lenstra 2002, Pomerance 2002, Berrizbeitia 2002, Cheng 2003, Bernstein 2003a/b, Lenstra and Pomerance 2003), which improved the speed of computation greatly.\nOwing to the existence of the many variants, Crandall and Papadopoulos refer to the \"AKS-class\" of algorithms in their scientific paper \"On the implementation of AKS-class primality tests\", published in March 2003.\nIn 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.\n(This version was eventually published in [Annals of Mathematics](https://bliptext.com/articles/annals-of-mathematics).) While the basic idea remained the same, r was chosen in a new manner, and the proof of correctness was more coherently organized.\nThe new proof relied almost exclusively on the behavior of [cyclotomic polynomials](https://bliptext.com/articles/cyclotomic-polynomial) over [finite fields](https://bliptext.com/articles/finite-fields).\nThe new upper bound on time complexity was, later reduced using additional results from [sieve theory](https://bliptext.com/articles/sieve-theory) to.\nIn 2005, [Pomerance](https://bliptext.com/articles/carl-pomerance) and [Lenstra](https://bliptext.com/articles/hendrik-lenstra) demonstrated a variant of AKS that runs in operations, leading to another updated version of the paper.\nAgrawal, Kayal and Saxena proposed a variant which would run in if [Agrawal's conjecture](https://bliptext.com/articles/agrawal-s-conjecture) were true; however, a heuristic argument by Pomerance and Lenstra suggested that it is probably false.\n\n# The algorithm\nThe algorithm is as follows:\nn > 1\n.\nn = ab\nfor integers\na > 1\nand\nb > 1\n, then output composite.\nordr(n) > (log2 n)2\n. If r and n are not coprime, then output composite.\na = 1\nto do\nHere [ordr](https://bliptext.com/articles/multiplicative-order)(n) is the [multiplicative order](https://bliptext.com/articles/multiplicative-order) of n [modulo](https://bliptext.com/articles/modular-arithmetic) r, log2 is the [binary logarithm](https://bliptext.com/articles/binary-logarithm), and \\varphi(r) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) of r.\nStep 3 is shown in the paper as checking 1 < (a,n) < n for all a ≤ r.\nIt can be seen this is equivalent to trial division up to r, which can be done very efficiently without using [gcd](https://bliptext.com/articles/greatest-common-divisor).\nSimilarly the comparison in step 4 can be replaced by having the trial division return prime once it has checked all values up to and including\nOnce beyond very small inputs, step 5 dominates the time taken.\nThe essential reduction in complexity (from exponential to polynomial) is achieved by performing all calculations in the finite ring\nconsisting of n^r elements.\nThis ring contains only the r [monomials](https://bliptext.com/articles/monomial), and the [coefficients](https://bliptext.com/articles/coefficient) are in which has n elements, all of them codable within \\log_2(n) bits.\nMost later improvements made to the algorithm have concentrated on reducing the size of r, which makes the core operation in step 5 faster, and in reducing the size of s, the number of loops performed in step 5.\nTypically these changes do not change the computational complexity, but can lead to many orders of magnitude less time taken; for example, Bernstein's final version has a theoretical speedup by a factor of over 2 million.\n\n## Proof of validity outline\nFor the algorithm to be correct, all steps that identify n must be correct.\nSteps 1, 3, and 4 are trivially correct, since they are based on direct tests of the divisibility of n.\nStep 5 is also correct: since (2) is true for any choice of a coprime to n and r if n is prime, an inequality means that n must be composite.\nThe difficult part of the proof is showing that step 6 is true.\nIts proof of correctness is based on the upper and lower bounds of a [multiplicative group](https://bliptext.com/articles/multiplicative-group) in constructed from the (X + a) binomials that are tested in step 5.\nStep 4 guarantees that these binomials are distinct elements of.\nFor the particular choice of r, the bounds produce a [contradiction](https://bliptext.com/articles/proof-by-contradiction) unless n is prime or a power of a prime.\nTogether with the test of step 1, this implies that n is always prime at step 6.\n\n## Example 1: n = 31 is prime\n{{pre|style=white-space:pre;overflow:auto;\n**Input**: integer n = 31 > 1.\n(* Step 1 *)\n**If** (n = a{{sup|b}} for integers a > 1 and b > 1),\n**output** composite.\n**For** ( b = 2; b <= log{{sub|2}}(n); b++) {\na = n{{sup|1/b}};\n**If** (a is integer),\n**Return**[Composite]\n} a = n{{sup|1/2}}...n{{sup|1/4}} = {5.568, 3.141, 2.360}\n(* Step 2 *)\nFind the smallest r such that O{{sub|r}}(n) > (log{{sub|2}} n){{sup|2}}.\nmaxk = ⌊(log{{sub|2}} n){{sup|2}}⌋;\nmaxr = Max[3, ⌈(Log{{sub|2}} n){{sup|5}}⌉]; (* maxr really isn't needed *)\nnextR = True;\n**For** (r = 2; nextR && r < maxr; r++) {\nnextR = False;\n**For** (k = 1; (!nextR) && k ≤ maxk; k++) {\nnextR = (Mod[n{{sup|k}}, r] == 1 {{!!}} Mod[n{{sup|k}}, r]==0)\n} } r--; (*the loop over increments by one*)\nr = 29\n(* Step 3 *)\n**If** (1 < [gcd](https://bliptext.com/articles/greatest-common-divisor)(a,n) < n for some a ≤ r),\n**output** composite.\n**For** (a = r; a > 1; a--) {\n**If** ((gcd = GCD[a,n]) > 1 && gcd < n),\n**Return**[Composite]\n} gcd = {GCD(29,31)=1, GCD(28,31)=1, ..., GCD(2,31)=1} ≯ 1\n(* Step 4 *)\n**If** (n ≤ r),\n**output** prime.\n**If** (n ≤ r),\n**Return**[Prime] (* this step may be omitted if n > 5690034 *)\n31 > 29\n(* Step 5 *)\n**For** a = 1 **to** **do**\n**If** ((X+a){{sup|n}} ≠ X{{sup|n}} + a (mod X{{sup|r}} − 1,n)),\n**output** composite;\nφ[x_] := EulerPhi[x];\nPolyModulo[f_] := PolynomialMod [ [PolynomialRemainder](https://bliptext.com/articles/polynomial-remainder)[f, x{{sup|r}}-1, x], n];\nmax = Floor[Log[2, n]{{sqrt|φ[r]}}];\n**For** (a = 1; a ≤ max; a++) {\n**If** (PolyModulo[(x+a){{sup|n}} - PolynomialRemainder[x{{sup|n}}+a, x{{sup|r}}-1], x] ≠ 0) {\n**Return**[Composite]\n{ } (x+a){{sup|31}} =\na{{sup|31}} +31a{{sup|30}}x +465a{{sup|29}}x{{sup|2}} +4495a{{sup|28}}x{{sup|3}} +31465a{{sup|27}}x{{sup|4}} +169911a{{sup|26}}x{{sup|5}} +736281a{{sup|25}}x{{sup|6}} +2629575a{{sup|24}}x{{sup|7}} +7888725a{{sup|23}}x{{sup|8}} +20160075a{{sup|22}}x{{sup|9}} +44352165a{{sup|21}}x{{sup|10}} +84672315a{{sup|20}}x{{sup|11}} +141120525a{{sup|19}}x{{sup|12}} +206253075a{{sup|18}}x{{sup|13}} +265182525a{{sup|17}}x{{sup|14}} +300540195a{{sup|16}}x{{sup|15}} +300540195a{{sup|15}}x{{sup|16}} +265182525a{{sup|14}}x{{sup|17}} +206253075a{{sup|13}}x{{sup|18}} +141120525a{{sup|12}}x{{sup|19}} +84672315a{{sup|11}}x{{sup|20}} +44352165a{{sup|10}}x{{sup|21}} +20160075a{{sup|9}}x{{sup|22}} +7888725a{{sup|8}}x{{sup|23}} +2629575a{{sup|7}}x{{sup|24}} +736281a{{sup|6}}x{{sup|25}} +169911a{{sup|5}}x{{sup|26}} +31465a{{sup|4}}x{{sup|27}} +4495a{{sup|3}}x{{sup|28}} +465a{{sup|2}}x{{sup|29}} +31ax{{sup|30}} +x{{sup|31}}\nPolynomialRemainder [(x+a){{sup|31}}, x{{sup|29}}-1] =\n465a{{sup|2}} +a{{sup|31}} +(31a+31a{{sup|30}})x +(1+465a{{sup|29}})x{{sup|2}} +4495a{{sup|28}}x{{sup|3}} +31465a{{sup|27}}x{{sup|4}} +169911a{{sup|26}}x{{sup|5}} +736281a{{sup|25}}x{{sup|6}} +2629575a{{sup|24}}x{{sup|7}} +7888725a{{sup|23}}x{{sup|8}} +20160075a{{sup|22}}x{{sup|9}} +44352165a{{sup|21}}x{{sup|10}} +84672315a{{sup|20}}x{{sup|11}} +141120525a{{sup|19}}x{{sup|12}} +206253075a{{sup|18}}x{{sup|13}} +265182525a{{sup|17}}x{{sup|14}} +300540195a{{sup|16}}x{{sup|15}} +300540195a{{sup|15}}x{{sup|16}} +265182525a{{sup|14}}x{{sup|17}} +206253075a{{sup|13}}x{{sup|18}} +141120525a{{sup|12}}x{{sup|19}} +84672315a{{sup|11}}x{{sup|20}} +44352165a{{sup|10}}x{{sup|21}} +20160075a{{sup|9}}x{{sup|22}} +7888725a{{sup|8}}x{{sup|23}} +2629575a{{sup|7}}x{{sup|24}} +736281a{{sup|6}}x{{sup|25}} +169911a{{sup|5}}x{{sup|26}} +31465a{{sup|4}}x{{sup|27}} +4495a{{sup|3}}x{{sup|28}}\n({{EquationRef|A}}) PolynomialMod [PolynomialRemainder [(x+a){{sup|31}}, x{{sup|29}}-1], 31] = a{{sup|31}}+x{{sup|2}}\n({{EquationRef|B}}) PolynomialRemainder [x{{sup|31}}+a, x{{sup|29}}-1] = a+x{{sup|2}}\n({{EquationNote|A}}) - ({{EquationNote|B}}) = a{{sup|31}}+x{{sup|2}} - (a+x{{sup|2}}) = a{{sup|31}}-a\n{1{{sup|31}}-1 = 0 (mod 31), 2{{sup|31}}-2 = 0 (mod 31), 3{{sup|31}}-3 = 0 (mod 31), ..., 26{{sup|31}}-26 = 0 (mod 31)}\n(* Step 6 *)\n**Output** prime.\n{{samp|31 Must be Prime}}\n}} Where PolynomialMod is a term-wise modulo reduction of the polynomial.\ne.g. PolynomialMod[x+2x{{sup|2}}+3x{{sup|3}}, 3] = x+2x{{sup|2}}+0x{{sup|3}}","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T19:28:55.414Z","updated_at":"2025-01-25T19:28:55.414Z","search_vector":"'ak':1 'primal':2 'test':3","search_content":" \n \n\n# \nThe **AKS primality test** (also known as **Agrawal–Kayal–Saxena primality test** and **cyclotomic AKS test**) is a [deterministic](https://bliptext.com/articles/deterministic-[algorithm](https://bliptext.com/articles/algorithm)) [primality-proving](https://bliptext.com/articles/primality-test) [algorithm](https://bliptext.com/articles/algorithm) created and published by [Manindra Agrawal](https://bliptext.com/articles/manindra-agrawal), [Neeraj Kayal](https://bliptext.com/articles/neeraj-kayal), and [Nitin Saxena](https://bliptext.com/articles/nitin-saxena), computer scientists at the [Indian Institute of Technology Kanpur](https://bliptext.com/articles/indian-institute-of-technology-kanpur), on August 6, 2002, in an article titled \"PRIMES is in P\".\nThe algorithm was the first one which is able to determine in [polynomial time](https://bliptext.com/articles/polynomial-time), whether a given number is [prime](https://bliptext.com/articles/prime-number) or [composite](https://bliptext.com/articles/composite-number) without relying on [mathematical conjectures](https://bliptext.com/articles/mathematical-conjecture) such as the [generalized Riemann hypothesis](https://bliptext.com/articles/generalized-riemann-hypothesis).\nThe proof is also notable for not relying on the field of [analysis](https://bliptext.com/articles/analysis-mathematics).\nIn 2006 the authors received both the [Gödel Prize](https://bliptext.com/articles/g-del-prize) and [Fulkerson Prize](https://bliptext.com/articles/fulkerson-prize) for their work.\n\n# Importance\nAKS is the first primality-proving algorithm to be simultaneously general, polynomial-time, deterministic, and unconditionally correct.\nPrevious algorithms had been developed for centuries and achieved three of these properties at most, but not all four.\nWhile the algorithm is of immense theoretical importance, it is not used in practice, rendering it a [galactic algorithm](https://bliptext.com/articles/galactic-algorithm).\nFor 64-bit inputs, the [Baillie–PSW test](https://bliptext.com/articles/baillie-psw-primality-test) is deterministic and runs many orders of magnitude faster.\nFor larger inputs, the performance of the (also unconditionally correct) ECPP and APR tests is far superior to AKS.\nAdditionally, ECPP can output a [primality certificate](https://bliptext.com/articles/primality-certificate) that allows independent and rapid verification of the results, which is not possible with the AKS algorithm.\n\n# Concepts\nThe AKS primality test is based upon the following theorem: Given an integer n\\ge 2 and integer a [coprime](https://bliptext.com/articles/coprime) to n, n is prime if and only if the [polynomial](https://bliptext.com/articles/polynomial-ring) [congruence relation](https://bliptext.com/articles/congruence-relation)\nholds within the polynomial ring.\nNote that X denotes the [indeterminate](https://bliptext.com/articles/polynomial-ring) which generates this polynomial ring.\nThis theorem is a generalization to polynomials of [Fermat's little theorem](https://bliptext.com/articles/fermat-s-little-theorem).\nIn one direction it can easily be proven using the [binomial theorem](https://bliptext.com/articles/binomial-theorem) together with the following property of the [binomial coefficient](https://bliptext.com/articles/binomial-coefficient):\nWhile the relation constitutes a primality test in itself, verifying it takes [exponential time](https://bliptext.com/articles/exponential-time): the [brute force](https://bliptext.com/articles/brute-force-method) approach would require the expansion of the (X + a)^n polynomial and a reduction \\pmod{n} of the resulting n + 1 coefficients.\nThe congruence is an equality in the [polynomial ring](https://bliptext.com/articles/polynomial-ring).\nEvaluating in a quotient ring of creates an upper bound for the degree of the polynomials involved.\nThe AKS evaluates the equality in, making the [computational complexity](https://bliptext.com/articles/computational-complexity-theory) dependent on the size of r. For clarity, this is expressed as the congruence\nwhich is the same as:\nfor some polynomials f and g.\nNote that all primes satisfy this relation (choosing g=0 in gives, which holds for n prime).\nThis congruence can be checked in polynomial time when r is polynomial to the digits of n. The AKS algorithm evaluates this congruence for a large set of a values, whose size is polynomial to the digits of n. The proof of validity of the AKS algorithm shows that one can find an r and a set of a values with the above properties such that if the congruences hold then n is a power of a prime.\n\n# History and running time\nIn the first version of the above-cited paper, the authors proved the asymptotic time complexity of the algorithm to be (using [Õ](https://bliptext.com/articles/big-o-notation) from [big O notation](https://bliptext.com/articles/big-o-notation))—the twelfth power of the number of digits in n times a factor that is polylogarithmic in the number of digits.\nHowever, this upper bound was rather loose; a widely-held conjecture about the distribution of the [Sophie Germain primes](https://bliptext.com/articles/sophie-germain-prime) would, if true, immediately cut the worst case down to.\nIn the months following the discovery, new variants appeared (Lenstra 2002, Pomerance 2002, Berrizbeitia 2002, Cheng 2003, Bernstein 2003a/b, Lenstra and Pomerance 2003), which improved the speed of computation greatly.\nOwing to the existence of the many variants, Crandall and Papadopoulos refer to the \"AKS-class\" of algorithms in their scientific paper \"On the implementation of AKS-class primality tests\", published in March 2003.\nIn 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.\n(This version was eventually published in [Annals of Mathematics](https://bliptext.com/articles/annals-of-mathematics).) While the basic idea remained the same, r was chosen in a new manner, and the proof of correctness was more coherently organized.\nThe new proof relied almost exclusively on the behavior of [cyclotomic polynomials](https://bliptext.com/articles/cyclotomic-polynomial) over [finite fields](https://bliptext.com/articles/finite-fields).\nThe new upper bound on time complexity was, later reduced using additional results from [sieve theory](https://bliptext.com/articles/sieve-theory) to.\nIn 2005, [Pomerance](https://bliptext.com/articles/carl-pomerance) and [Lenstra](https://bliptext.com/articles/hendrik-lenstra) demonstrated a variant of AKS that runs in operations, leading to another updated version of the paper.\nAgrawal, Kayal and Saxena proposed a variant which would run in if [Agrawal's conjecture](https://bliptext.com/articles/agrawal-s-conjecture) were true; however, a heuristic argument by Pomerance and Lenstra suggested that it is probably false.\n\n# The algorithm\nThe algorithm is as follows:\nn > 1\n.\nn = ab\nfor integers\na > 1\nand\nb > 1\n, then output composite.\nordr(n) > (log2 n)2\n. If r and n are not coprime, then output composite.\na = 1\nto do\nHere [ordr](https://bliptext.com/articles/multiplicative-order)(n) is the [multiplicative order](https://bliptext.com/articles/multiplicative-order) of n [modulo](https://bliptext.com/articles/modular-arithmetic) r, log2 is the [binary logarithm](https://bliptext.com/articles/binary-logarithm), and \\varphi(r) is [Euler's totient function](https://bliptext.com/articles/euler-s-totient-function) of r.\nStep 3 is shown in the paper as checking 1 1.\n(* Step 1 *)\n**If** (n = a{{sup|b}} for integers a > 1 and b > 1),\n**output** composite.\n**For** ( b = 2; b (log{{sub|2}} n){{sup|2}}.\nmaxk = ⌊(log{{sub|2}} n){{sup|2}}⌋;\nmaxr = Max[3, ⌈(Log{{sub|2}} n){{sup|5}}⌉]; (* maxr really isn't needed *)\nnextR = True;\n**For** (r = 2; nextR && r < maxr; r++) {\nnextR = False;\n**For** (k = 1; (!nextR) && k ≤ maxk; k++) {\nnextR = (Mod[n{{sup|k}}, r] == 1 {{!!}} Mod[n{{sup|k}}, r]==0)\n} } r--; (*the loop over increments by one*)\nr = 29\n(* Step 3 *)\n**If** (1 < [gcd](https://bliptext.com/articles/greatest-common-divisor)(a,n) < n for some a ≤ r),\n**output** composite.\n**For** (a = r; a > 1; a--) {\n**If** ((gcd = GCD[a,n]) > 1 && gcd < n),\n**Return**[Composite]\n} gcd = {GCD(29,31)=1, GCD(28,31)=1, ..., GCD(2,31)=1} ≯ 1\n(* Step 4 *)\n**If** (n ≤ r),\n**output** prime.\n**If** (n ≤ r),\n**Return**[Prime] (* this step may be omitted if n > 5690034 *)\n31 > 29\n(* Step 5 *)\n**For** a = 1 **to** **do**\n**If** ((X+a){{sup|n}} ≠ X{{sup|n}} + a (mod X{{sup|r}} − 1,n)),\n**output** composite;\nφ[x_] := EulerPhi[x];\nPolyModulo[f_] := PolynomialMod [ [PolynomialRemainder](https://bliptext.com/articles/polynomial-remainder)[f, x{{sup|r}}-1, x], n];\nmax = Floor[Log[2, n]{{sqrt|φ[r]}}];\n**For** (a = 1; a ≤ max; a++) {\n**If** (PolyModulo[(x+a){{sup|n}} - PolynomialRemainder[x{{sup|n}}+a, x{{sup|r}}-1], x] ≠ 0) {\n**Return**[Composite]\n{ } (x+a){{sup|31}} =\na{{sup|31}} +31a{{sup|30}}x +465a{{sup|29}}x{{sup|2}} +4495a{{sup|28}}x{{sup|3}} +31465a{{sup|27}}x{{sup|4}} +169911a{{sup|26}}x{{sup|5}} +736281a{{sup|25}}x{{sup|6}} +2629575a{{sup|24}}x{{sup|7}} +7888725a{{sup|23}}x{{sup|8}} +20160075a{{sup|22}}x{{sup|9}} +44352165a{{sup|21}}x{{sup|10}} +84672315a{{sup|20}}x{{sup|11}} +141120525a{{sup|19}}x{{sup|12}} +206253075a{{sup|18}}x{{sup|13}} +265182525a{{sup|17}}x{{sup|14}} +300540195a{{sup|16}}x{{sup|15}} +300540195a{{sup|15}}x{{sup|16}} +265182525a{{sup|14}}x{{sup|17}} +206253075a{{sup|13}}x{{sup|18}} +141120525a{{sup|12}}x{{sup|19}} +84672315a{{sup|11}}x{{sup|20}} +44352165a{{sup|10}}x{{sup|21}} +20160075a{{sup|9}}x{{sup|22}} +7888725a{{sup|8}}x{{sup|23}} +2629575a{{sup|7}}x{{sup|24}} +736281a{{sup|6}}x{{sup|25}} +169911a{{sup|5}}x{{sup|26}} +31465a{{sup|4}}x{{sup|27}} +4495a{{sup|3}}x{{sup|28}} +465a{{sup|2}}x{{sup|29}} +31ax{{sup|30}} +x{{sup|31}}\nPolynomialRemainder [(x+a){{sup|31}}, x{{sup|29}}-1] =\n465a{{sup|2}} +a{{sup|31}} +(31a+31a{{sup|30}})x +(1+465a{{sup|29}})x{{sup|2}} +4495a{{sup|28}}x{{sup|3}} +31465a{{sup|27}}x{{sup|4}} +169911a{{sup|26}}x{{sup|5}} +736281a{{sup|25}}x{{sup|6}} +2629575a{{sup|24}}x{{sup|7}} +7888725a{{sup|23}}x{{sup|8}} +20160075a{{sup|22}}x{{sup|9}} +44352165a{{sup|21}}x{{sup|10}} +84672315a{{sup|20}}x{{sup|11}} +141120525a{{sup|19}}x{{sup|12}} +206253075a{{sup|18}}x{{sup|13}} +265182525a{{sup|17}}x{{sup|14}} +300540195a{{sup|16}}x{{sup|15}} +300540195a{{sup|15}}x{{sup|16}} +265182525a{{sup|14}}x{{sup|17}} +206253075a{{sup|13}}x{{sup|18}} +141120525a{{sup|12}}x{{sup|19}} +84672315a{{sup|11}}x{{sup|20}} +44352165a{{sup|10}}x{{sup|21}} +20160075a{{sup|9}}x{{sup|22}} +7888725a{{sup|8}}x{{sup|23}} +2629575a{{sup|7}}x{{sup|24}} +736281a{{sup|6}}x{{sup|25}} +169911a{{sup|5}}x{{sup|26}} +31465a{{sup|4}}x{{sup|27}} +4495a{{sup|3}}x{{sup|28}}\n({{EquationRef|A}}) PolynomialMod [PolynomialRemainder [(x+a){{sup|31}}, x{{sup|29}}-1], 31] = a{{sup|31}}+x{{sup|2}}\n({{EquationRef|B}}) PolynomialRemainder [x{{sup|31}}+a, x{{sup|29}}-1] = a+x{{sup|2}}\n({{EquationNote|A}}) - ({{EquationNote|B}}) = a{{sup|31}}+x{{sup|2}} - (a+x{{sup|2}}) = a{{sup|31}}-a\n{1{{sup|31}}-1 = 0 (mod 31), 2{{sup|31}}-2 = 0 (mod 31), 3{{sup|31}}-3 = 0 (mod 31), ..., 26{{sup|31}}-26 = 0 (mod 31)}\n(* Step 6 *)\n**Output** prime.\n{{samp|31 Must be Prime}}\n}} Where PolynomialMod is a term-wise modulo reduction of the polynomial.\ne.g. PolynomialMod[x+2x{{sup|2}}+3x{{sup|3}}, 3] = x+2x{{sup|2}}+0x{{sup|3}}","revisionCount":0},"hasMore":false,"nextCursor":null}