MZd 6ddlmZmZmZddlmZdZdZdZy))IntegerPowMod) factorintc d|dkst||k7rtd|zt|}tt |j }d}|D]U\}}|D]F\}}t td|dzDcgc]}tt|||dk(c}sDd}n|rT|S|Scc}w)aJ Check whether `n` is a nilpotent number. A number `n` is said to be nilpotent if and only if every finite group of order `n` is nilpotent. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_nilpotent_number >>> from sympy import randprime >>> is_nilpotent_number(21) False >>> is_nilpotent_number(randprime(1, 30)**12) True References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. r$n must be a positive integer, not %iTF) int ValueErrorrlistritemsanyrangerr)n prime_factors is_nilpotentp_ja_jp_ia_iks C/usr/lib/python3/dist-packages/sympy/combinatorics/group_numbers.pyis_nilpotent_numberrs0 AvQ1?!CDD A1++-.ML!S% HCaq8IJ1CC S)Q.JK$      Ks7 B- c|dkst||k7rtd|zt|}t|syt t |j }td|D}|S)af Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. rrFc3,K|] \}}|dkyw)N.0rrs r z$is_abelian_number..Ps;cS1W;r r rrr rr all)rr is_abelians ris_abelian_numberr%.sg4 AvQ1?!CDD A q !1++-.M;];;J c|dkst||k7rtd|zt|}t|syt t |j }td|D}|S)a^ Check whether `n` is a cyclic number. A number `n` is said to be cyclic if and only if every finite group of order `n` is cyclic. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_cyclic_number >>> from sympy import randprime >>> is_cyclic_number(15) True >>> is_cyclic_number(randprime(1, 2000)**2) False >>> is_cyclic_number(4) False References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. rrFc3,K|] \}}|dkyw)Nrrs rr z#is_cyclic_number..us:SC!G:r!r")rr is_cyclics ris_cyclic_numberr+Tsg2 AvQ1?!CDD A q !1++-.M:M::I r&N) sympy.corerrrsympyrrr%r+rr&rr.s((&R#L"r&