- Primitive root modulo n
In

modular arithmetic , a branch ofnumber theory , a**primitive root modulo "n**" is any number "g" with the property that any numbercoprime to "n" is congruent to a power of "g" (mod "n"). That is, if "g" is a primitive root (mod "n") and gcd("a", "n") = 1, then there is an integer "k" such that "g"^{"k"}≡ "a" (mod "n"). "k" is called the**index**of "a".Gauss defines primitive root in Article 57 of the

Disquisitiones Arithmeticae (1801), where he creditsEuler with coining the term. In Art. 56 he states that Lambert and Euler knew of them, but that his is the first rigorous demonstration that they exist. In fact, the "Disquisitiones" has two proofs: the one in Art. 54 is a nonconstructive existence proof, the one in Art. 55 is constructive.**Definition and examples**If "n" is a positive

integer thecongruence class escoprime to "n" form a group with multiplication modulo "n" as the operation; it is denoted by**Z**_{"n"}^{×}and is called the group of units (mod "n") or the group of primitive classes (mod "n"). As explained in the articlemultiplicative group of integers modulo n , this group is cyclicif and only if "n" is equal [*Gauss, DA, arts 82-92*] to 1, 2, 4, "p"^{"k"}, or 2 "p"^{"k"}where "p"^{"k"}is a power of an oddprime number . A generator of this cyclic group is called a**primitive root modulo "n**", or a**primitive element of Z**_{"n"}^{×}.The order of (i.e. the number of elements in)

**Z**_{"n"}^{×}is given byEuler's totient function $varphileft(n\; ight).$Euler's theorem says that "a"^{φ("n")}≡ 1 (mod "n") for every "a" coprime to "n"; the lowest power of "a" which is ≡ 1 (mod "n") is called themultiplicative order of "a" (mod "n"). In other words, for "a" to be a primitive root modulo "n", φ("n") has to be the smallest power of "a" which is ≡ 1 (mod "n").Take for example "n" = 14. The elements of

**Z**_{14}^{×}are the congruence classes {1, 3, 5, 9, 11, 13}; there are φ(14) = 6 of them.Here is a table of their powers (mod 14):

`n n, n`

^{2}, n^{3}, ... (mod 14) 1 : 1, 3 : 3, 9, 13, 11, 5, 1 5 : 5, 11, 13, 9, 3, 1 9 : 9, 11, 1 11 : 11, 9, 1 13 : 13, 1The order of 1 is 1, the orders of 3 and 5 are 6, the orders of 9 and 11 are 3, and the order of 13 is 2. Thus, 3 and 5 are the primitive roots modulo 14.

For a second example let "n" = 15. The elements of

**Z**_{15}^{×}are the congruence classes {1, 2, 4, 7, 8, 11, 13, 14}; there are φ(15) = 8 of them.`n n, n`

^{2}, n^{3}, ... (mod 15) 1 : 1 2 : 2, 4, 8, 1 4 : 4, 1 7 : 7, 4, 13, 1 8 : 8, 4, 2, 1 11 : 11, 1 13 : 13, 4, 7, 1 14 : 14, 1Since there is no number whose order is 8, there are no primitive roots (mod 15).

**Table of primitive roots**Given an appropropriate modulus "n", a primitive root "g" (mod "n") and a number "a" coprime to "n", the exponent to which "g" must be raised to be congruent to "a" (mod "n") is called the

**index of "a".**It resembles a logarithm, in that multiplication becomes the addition of indices and exponentiation becomes multiplication of indices. It is usually denoted by ν("a"), i.e. "g"^{ν("a")}≡ "a" (mod "n").This is Gauss's table of primitive roots from the "Disquisitiones". Unlike most modern authors he did not always choose the smallest primitive root. Instead, he chose 10 if it is a primitive root; if it isn't, he chose whichever root gives 10 the smallest index, and, if there is more than one, chose the smallest of them. This is not only to make hand calculation easier, but is used in § VI where the periodic decimal expansions of rational numbers are investigated.

The rows of the table are labeled with the prime powers (excepting 2, 4, and 8) less than 100; the second column is a primitive root modulo that number. The columns are labeled with the primes less than 97. The entry in row "p" column "q" is the index of "q" (mod "p") for the given root.

For example, in row 11, 2 is given as the primitive root, and in column 5 the entry is 4. This means that 2

For the index of a composite number, add the indices of its prime factors.^{4}= 16 ≡ 5 (mod 11).For example, in row 11, the index of 6 is the sum of the indices for 2 and 3: 2

^{1 + 8}= 512 ≡ 6 (mod 11). The index of 25 is twice the index 5: 2^{8}= 256 ≡ 25 (mod 11). (Of course, since 25 ≡ 3 (mod 11), the entry for 3 is 8).The table is straightforward for the odd prime powers. But the powers of 2, (16, 32, and 64) do not have primitive roots; instead, the powers of 5 account for one-half of the odd numbers less than the power of 2, and their negatives modulo the power of 2 account for the other half. All powers of 5 are ≡ 5 or 1 (mod 8); the columns headed by numbers ≡ 3 or 7 (mod 8) contain the index of its negative.

For example, (mod 32) the index for 7 is 2, and 5

^{2}= 25 ≡ –7 (mod 32), but the entry for 17 is 4, and 5^{4}= 625 ≡ 17 (mod 32).The sequence of smallest primitive roots may be found at OEIS|id=A046145.

**Arithmetic facts**Gauss proved [

*Gauss, DA, art. 80*] that for any prime number "p" (with the sole exception of 3) the product of its primitive roots is ≡ 1 (mod "p").He also proved [

*Gauss, DA, art. 81*] that for any prime number "p" the sum of its primitive roots is ≡ μ("p" – 1) (mod "p") where μ is theMöbius function .For example

:"p" = 3, μ(2) = –1. The primitive root is 2.

:"p" = 5, μ(4) = 0. The primitive roots are 2 and 3.

:"p" = 7, μ(6) = 1. The primitive roots are 3 and 5.

:"p" = 31, μ(30) = –1. The primitive roots are 3, 11, 12, 13, 17 ≡ –14, 21 ≡ –10, 22 ≡ –9, and 24 ≡ –7.

Their product 970377408 ≡ 1 (mod 31) and their sum 123 ≡ –1 (mod 31).

::3×11 = 33 ≡ 2::12×13 = 156 ≡ 1::(–14)×(–10) = 140 ≡ 16::(–9)×(–7) = 63 ≡ 1, and 2×1×16×1 = 32 ≡ 1 (mod 31).

**Finding primitive roots**No simple general formula to compute primitive roots modulo "n" is known. There are however methods to locate a primitive root that are faster than simply trying out all candidates. If the

multiplicative order of a number "m" modulo "n" is equal to $phileft(n\; ight)$ (the order of**Z**_{"n"}^{*}), then it is a primitive root. In fact the converse is true: If m is a primitive root modulo n then the multiplicative order of m is $phileft(n\; ight)$. We can use this to test for primitive roots.First, compute $phileft(n\; ight)$. Then determine the different prime factors of $phileft(n\; ight)$, say "p"

_{1},...,"p"_{"k"}. Now, for every element "m" of**Z**_{"n"}^{*}, compute :$m^\{phi(n)/p\_i\}mod\; n\; qquadmbox\{\; for\; \}\; i=1,ldots,k$using a fast algorithm formodular exponentiation such asexponentiation by squaring . A number "m" for which these "k" results are all different from 1 is a primitive root.The number of primitive roots modulo "n", if there are any, is equal to

:$phileft(phileft(n\; ight)\; ight)$

since, in general, a cyclic group with "r" elements has $phileft(r\; ight)$ generators.

If "g" is a primitive root (mod "p") then "g" is a primitive root modulo all powers "p"

^{"k"}unless "g"^{"p" – 1}≡ 1 (mod "p"^{2}); in that case, "g" + "p" is. [*This and the next assertion are in Cohen, p.26*]If "g" is a primitive root (mod "p"

^{"k"}), then "g" or "g" + "p"^{"k"}(whichever one is odd) is a primitive root (mod 2"p"^{"k"}).**Order of magnitude of primitive roots**The least primitive root (mod "p") is generally small.

Let "g"

_{"p"}be the smallest primitive root (mod "p") in the range 1, 2, ... "p"–1.Fridlander (1949) and Salié (1950) proved [

*Ribenboim, p.24*] that that there is a constant "C" such that for infinitely many primes "g"_{"p"}> "C" log "p".It can be proved [

*Ribenboim, p.24*] in an elementary manner that for any positive integer "M" there are infinitely many primes such that "M" < "g"_{"p"}< "p" – "M".Burgess (1962) proved [

*Ribenboim, p.24*] that for every ε > 0 there is a "C" such that $g\_p\; <\; Cp^\{frac\{1\}\{4\}+epsilon\}.$Grosswald (1981) proved [

*Ribenboim, p.24*] that $mbox\{\; if\; \}\; p\; >\; e^\{e^\{24\; mbox\{\; then\; \}g\_p\; <\; p^\{0.499\}$Shoup (1990, 1992) proved, [

*Bach & Shallit, p.254*] assuming thegeneralized Riemann hypothesis , that "g"_{"p"}=O(log^{6}"p").**Uses**Primitive root modulo n is often used in

cryptography , including theDiffie-Hellman Key Exchange Scheme.**ee also***

Artin's conjecture on primitive roots

*Dirichlet character

* Gauss' generalization of Wilson's theorem

* Order modulo n**Notes****References**The "

Disquisitiones Arithmeticae " has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.*citation

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last1 = Gauss | first1 = Carl Friedrich

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title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)

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title = Algorithmic Number Theory (Vol I: Efficient Algorithms)

publisher =The MIT Press

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date = 1966

isbn = 0-262-02045-5*citation

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title = A Course in Computational Algebraic Number Theory

publisher =Springer

location = Berlin

date = 1993

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authorlink=Øystein Ore

last = Ore

first = Oystein

title = Number Theory and Its History

publisher = Dover

date = 1988

pages = 284-302

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last1 = Ribenboim | first1 = Paulo

title = The New Book of Prime Number Records

publisher =Springer

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isbn = 0-387-94457-5**External links**This site has a [

*http://www.math.mtu.edu/mathlab/COURSES/holt/dnt/quadratic4.html primitive root calculator*] applet.

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