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单词 PrimitiveRoot
释义

primitive root


Given any positive integer n, the group of units U(/n) of the ring /n is a cyclic groupMathworldPlanetmath iff n is 4, pm or 2pm for any odd positive prime p and any non-negative integer m.  A primitive rootMathworldPlanetmath is a generatorPlanetmathPlanetmathPlanetmath of this group of units when it is cyclic.

Equivalently, one can define the integer r to be a primitive root modulo n, if the numbers r0,r1,,rn-2 form a reduced residue systemMathworldPlanetmath modulo n.

For example, 2 is a primitive root modulo 5, since1, 2, 22=4, 23=83(mod5)are all with 5 coprimeMathworldPlanetmathPlanetmath positive integers less than 5.

The generalized Riemann hypothesisMathworldPlanetmath implies that every prime numberMathworldPlanetmath p has a primitive root below 70(lnp)2.

References

Wikipedia, “Primitive root modulo n”

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