In number theory, a number is a primitive root modulo if every number coprime to is congruent to a power of modulo . In symbols, is a primitive root modulo if for every integer coprime to , there is some integer for which .
Primitive roots only exist for some integers . Specifically, a primitive root exists modulo if and only if is 4, or for some odd prime number and some . This was first proved by Carl Friedrich Gauss. Gauss defined primitive roots in Article 57 of his Disquisitiones Arithmeticae (1801), where he credited Leonhard Euler with coining the term. In Article 56 he stated that Johann Heinrich Lambert and Euler knew of them, but he was the first to rigorously demonstrate that primitive roots exist for a prime number . In fact, the Disquisitiones contains two proofs: The one in Article 54 is a nonconstructive existence proof, while the proof in Article 55 is constructive.
An equivalent characterization is that is a primitive root modulo if and only if is a generator of the multiplicative group of integers modulo. Thus, primitive roots exist if and only if this group is a cyclic group.
If is a primitive root modulo and , then the value is called the index or discrete logarithm of to the base modulo .
The number 3 is a primitive root modulo 7 because
The remainders 3, 2, 6, 4, 5, 1 include each congruence class relatively prime to 7. Higher powers repeat the same pattern periodically.
The number of congruence classes relatively prime to the modulus is given by Euler's totient function applied to . In this case, that is . For a prime modulus , this period is always equal to , but this is not true for composite .
If is a positive integer, the integers from 1 to that are coprime to (or equivalently, the congruence classes coprime to ) form a group, with multiplication modulo as the operation; it is denoted by , and is called the group of units modulo , or the group of primitive classes modulo . As explained in the article multiplicative group of integers modulo, this multiplicative group is cyclic if and only if is equal to 2, 4, , or where is a power of an odd prime number. When (and only when) this group is cyclic, a generator of this cyclic group is called a primitive root modulo (or in fuller language primitive root of unity modulo , emphasizing its role as a fundamental solution of the roots of unity polynomial equations X â 1 in the ring ), or simply a primitive element of .
When is non-cyclic, such primitive elements mod do not exist. Instead, each prime component of has its own sub-primitive roots (see in the examples below).
For any (whether or not is cyclic), the order of is given by Euler's totient function () . And then, Euler's theorem says that for every coprime to ; the lowest power of that is congruent to 1 modulo is called the multiplicative order of modulo . In particular, for to be a primitive root modulo , has to be the smallest power of satisfying .
For example, if then the elements of are the congruence classes {1, 3, 5, 9, 11, 13}; there are of them. Here is a table of their powers modulo 14:
x x, x<sup>2</sup>, x<sup>3</sup>, ... (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, 1
The 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 The elements of are the congruence classes {1, 2, 4, 7, 8, 11, 13, 14}; there are of them.
x x, x<sup>2</sup>, x<sup>3</sup>, ... (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, 1
Since there is no number whose order is 8, there are no primitive roots modulo 15. Indeed, , where is the Carmichael function.
Numbers that have a primitive root are of the shape
These are the numbers with kept also in the sequence in the OEIS.
The following table lists the primitive roots modulo up to :
Gauss proved that for any prime number (with the sole exception of the product of its primitive roots is congruent to 1 modulo .
He also proved that for any prime number , the sum of its primitive roots is congruent to ( â 1) modulo , where is the Möbius function.
For example,
E.g., the product of the latter primitive roots is , and their sum is .
If is a primitive root modulo the prime , then .
Artin's conjecture on primitive roots states that a given integer that is neither a perfect square nor −1 is a primitive root modulo infinitely many primes.
No simple general formula to compute primitive roots modulo is known. There are however methods to locate a primitive root that are faster than simply trying out all candidates. If the multiplicative order (its exponent) of a number modulo is equal to (the order of ), then it is a primitive root. In fact the converse is true: If is a primitive root modulo , then the multiplicative order of is We can use this to test a candidate to see if it is primitive.
For first, compute Then determine the different prime factors of , say <sub>1</sub>, ..., . Finally, compute
using a fast algorithm for modular exponentiation such as exponentiation by squaring. A number for which these results are all different from 1 is a primitive root.
The number of primitive roots modulo , if there are any, is equal to
since, in general, a cyclic group with elements has generators.
For prime , this equals , and since the generators are very common among {2, ..., −1} and thus it is relatively easy to find one.
If is a primitive root modulo , then is also a primitive root modulo all powers unless <sup>âÂÂ1</sup> â¡ 1 (mod <sup>2</sup>); in that case, + is.
If is a primitive root modulo , then is also a primitive root modulo all smaller powers of .
If is a primitive root modulo , then either or + (whichever one is odd) is a primitive root modulo 2.
Finding primitive roots modulo is also equivalent to finding the roots of the ( â 1)st cyclotomic polynomial modulo .
The least primitive root modulo (in the range 1, 2, ..., is generally small.
Burgess (1962) proved that for every õ > 0 there is a such that
Grosswald (1981) proved that if , then
Shoup (1990, 1992) proved, assuming the generalized Riemann hypothesis, that
Fridlander (1949) and Salié (1950) proved that there is a positive constant such that for infinitely many primes
It can be proved in an elementary manner that for any positive integer there are infinitely many primes such that < <
A primitive root modulo is often used in pseudorandom number generators and cryptography, including the DiffieâÂÂHellman key exchange scheme. Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.
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.