U —9%eã@s<ddlmZmZmZddlmZdd„Zdd„Zdd„Zd S) é)ÚIntegerÚPowÚMod)Ú factorintcsŽ|dkst|ƒ|kr td|ƒ‚t|ƒ}tt|ƒ ¡ƒ}d}|D]H\‰}|D]2\‰}t‡‡fdd„td|dƒDƒƒrLd}q€qL|s@qŠq@|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. rú$n must be a positive integer, not %iTcs g|]}ttˆ|ƒˆƒdk‘qS)é)rr)Ú.0Úk©Úp_iZp_j©ú`/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/combinatorics/group_numbers.pyÚ %sz'is_nilpotent_number..rF)ÚintÚ ValueErrorrÚlistrÚitemsÚanyÚrange)ÚnÚ prime_factorsZis_nilpotentZa_jÚa_irr r Úis_nilpotent_numbers"rcCsZ|dkst|ƒ|kr td|ƒ‚t|ƒ}t|ƒs4dStt|ƒ ¡ƒ}tdd„|Dƒƒ}|S)af Check whether `n` is an abelian number. A number `n` is said to be abelian if and only if every finite group of order `n` is abelian. For more information see [1]_. Examples ======== >>> from sympy.combinatorics.group_numbers import is_abelian_number >>> from sympy import randprime >>> is_abelian_number(4) True >>> is_abelian_number(randprime(1, 2000)**2) True >>> is_abelian_number(60) False References ========== .. [1] Pakianathan, J., Shankar, K., *Nilpotent Numbers*, The American Mathematical Monthly, 107(7), 631-634. rrFcss|]\}}|dkVqdS)éNr©rrrrrr Ú Psz$is_abelian_number..©rrrrrrrÚall)rrZ is_abelianrrr Úis_abelian_number.srcCsZ|dkst|ƒ|kr td|ƒ‚t|ƒ}t|ƒs4dStt|ƒ ¡ƒ}tdd„|Dƒƒ}|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. rrFcss|]\}}|dkVqdS)éNrrrrr rusz#is_cyclic_number..r)rrZ is_cyclicrrr Úis_cyclic_numberTsr N) Z sympy.corerrrZsympyrrrr rrrr Ús)&