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Prime Number Integer Ring Implication Biconditional Even Number Positive Cyclic Group Coprime Subalgebra Of An Algebraic System Presentationgroup Group Of Units Generalized Riemann Hypothesis Residue Systems

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Primitive Root

Given any positive integer , the group of units $U(\mathbb{Z}/n\mathbb{Z})$ of the ring $\mathbb{Z}/n\mathbb{Z}$ is a cyclic group iff is 4, or for any odd positive prime and any non-negative integer . A primitive root is a generator of this group of units when it is cyclic.

Equivalently, one can define the integer to be a primitive root modulo , if the numbers $r^0,\,r^1,\,\ldots,\,r^{n-2}$ form a reduced residue system modulo .

For example, 2 is a primitive root modulo 5, since $1,\; 2,\; 2^2 = 4,\; 2^3 = 8 \equiv 3 \pmod{5}$ are all with 5 coprime positive integers less than 5.

The generalized Riemann hypothesis implies that every prime number has a primitive root below $70(\ln p)^2$ .

Bibliography

Wikipedia, ``Primitive root modulo n''

1	Positive
1	Even Number
1	Integer
1	Biconditional
1	Number
1	Implication
1	Ring
1	Prime
2	Subalgebra Of An Algebraic System
2	Coprime
2	Cyclic Group
3	Presentationgroup
6	Group Of Units
10	Residue Systems
12	Generalized Riemann Hypothesis