Rie 8ddlmZmZmZddlmZdZdZdZdS))IntegerPowMod) factorintc h|dkst||krtd|zt|}tt |}d}|D]B\}|D]6\}t fdtd|dzDrd}n7|snC|S)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$n must be a positive integer, not %iTcTg|]$}tt|dk%S))rr).0kp_ip_js Alib/python3.11/site-packages/sympy/combinatorics/group_numbers.py z'is_nilpotent_number..%s0JJJ1CC S))Q.JJJr F)int ValueErrorrlistritemsanyrange)n prime_factors is_nilpotenta_ja_ir rs @@ris_nilpotent_numberrs0 AvvQ1?!CDDD A1++--..ML!S%  HCJJJJJaq8I8IJJJKK $    E  rc*|dkst||krtd|zt|}t|sdSt t |}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|] \}}|dkVdS)Nr r rs r z$is_abelian_number..Ps*;;cS1W;;;;;;rrrrrrrrall)rr is_abelians ris_abelian_numberr'.s4 AvvQ1?!CDDD A q ! !u1++--..M;;];;;;;J rc*|dkst||krtd|zt|}t|sdSt t |}td|D}|S)a^ Check whether `n` is a cyclic number. A number `n` is said to be cyclic if and only if every finite group of order `n` is cyclic. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_cyclic_number >>> from sympy import randprime >>> is_cyclic_number(15) True >>> is_cyclic_number(randprime(1, 2000)**2) False >>> is_cyclic_number(4) False References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. rrFc3(K|] \}}|dkVdS)Nr!r"s rr#z#is_cyclic_number..us*::SC!G::::::rr$)rr is_cyclics ris_cyclic_numberr,Ts2 AvvQ1?!CDDD A q ! !u1++--..M::M:::::I rN) sympy.corerrrsympyrrr'r,r!rrr/sn((((((((((&&&R###L"""""r