o *ήc @s<ddlmZmZmZddlmZddZddZddZd S) )IntegerPowMod) factorintcs|dks t||krtd|t|}tt|}d}|D]%\}|D]\}tfddtd|dDr?d}nq&|sE|Sq |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$n must be a positive integer, not %iTcs g|] }tt|dkqS))rr).0kp_ip_jH/tmp/pip-target-vg8gfxp4/lib/python/sympy/combinatorics/group_numbers.py %s z'is_nilpotent_number..rF)int ValueErrorrlistritemsanyrange)n prime_factors is_nilpotenta_ja_ir r ris_nilpotent_numbers    "rcCZ|dks t||krtd|t|}t|sdStt|}tdd|D}|S)af Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. rrFcs|] \}}|dkVqdS)Nr rr rr r r Pz$is_abelian_number..rrrrrrrall)rr is_abelianr r ris_abelian_number.s r%cCr)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. rrFcsr)Nr rr r rr ur!z#is_cyclic_number..r")rr is_cyclicr r ris_cyclic_numberTs r(N) sympy.corerrrsympyrrr%r(r r r rs  ) &