15.7 GeneratorsPrimeResidues

GeneratorsPrimeResidues( n )

returns a record with fields

primes:

the set of prime divisors of the integer n,

exponents:

the corresponding exponents in the factorization of n and

generators:

generators of the group of prime residues: For each odd prime p there is one generator, corresponding to a primitive root of the subgroup (Z/p^{nu_p})^{ast} of (Z/nZ)^{ast}, where nu_p is the exponent of p in the factorization of n; for p = 2, we have one generator in the case that 8 does not divide n, and a list of two generators (corresponding to = (Z/2^{nu_2})^{ast}) else.

    gap> GeneratorsPrimeResidues( 9 );      # 2 is a primitive root
    rec(
      primes := [ 3 ],
      exponents := [ 2 ],
      generators := [ 2 ] )
    gap> GeneratorsPrimeResidues( 24 );     # 8 divides 24
    rec(
      primes := [ 2, 3 ],
      exponents := [ 3, 1 ],
      generators := [ [ 7, 13 ], 17 ] )
    gap> GeneratorsPrimeResidues( 1155 );
    rec(
      primes := [ 3, 5, 7, 11 ],
      exponents := [ 1, 1, 1, 1 ],
      generators := [ 386, 232, 661, 211 ] )

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