
    \h
                        S r SSKJrJrJrJrJrJrJrJ	r	J
r
JrJrJr  SSKJrJrJr  SSKJrJrJrJrJrJrJrJrJrJrJrJrJrJ r J!r!J"r"J#r#J$r$J%r%J&r&J'r'J(r(J)r)J*r*J+r+J,r,J-r-  SSK.J/r/  SSK0J1r1J2r2J3r3J4r4J5r5J6r6J7r7J8r8J9r9J:r:J;r;J<r<J=r=J>r>J?r?  SSK@JArAJBrBJCrC  SSKDJErEJFrFJGrGJHrHJDrD  SS	KIJJrJJIrIJKrK  SS
KLJLrL  SSKMJMrM  SSKNJNrNJOrO  / SQrPg)z$
Number theory module (primes, etc)
   )	nextprime	prevprimeprimeprimepi
primerange	randprimeSievesieve	primorialcycle_length	compositecompositepi)isprimeis_gaussian_primeis_mersenne_prime)divisorsproper_divisors	factorintmultiplicitymultiplicity_in_factorialperfect_powerfactor_cachepollard_pm1pollard_rhoprimefactorstotientdivisor_countproper_divisor_countdivisor_sigma	factorratreduced_totientprimenu
primeomegamersenne_prime_exponent
is_perfectis_abundantis_deficientis_amicableis_carmichael	abundancedradrm)npartitions)is_primitive_rootis_quad_residuelegendre_symboljacobi_symboln_ordersqrt_modquadratic_residuesprimitive_rootnthroot_modis_nthpow_residuesqrt_mod_itermobiusdiscrete_logquadratic_congruencepolynomial_congruence)binomial_coefficientsbinomial_coefficients_listmultinomial_coefficients)continued_fraction_periodiccontinued_fraction_iteratorcontinued_fraction_reducecontinued_fraction_convergentscontinued_fraction)count_digitsdigitsis_palindromic)egyptian_fraction)ecm)qs	qs_factor)Ir   r   r   r   r   r   r	   r
   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   r   r-   r.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   r=   r>   r?   r@   rA   rB   rC   rD   rF   rE   rG   rH   rI   rJ   rK   N)Q__doc__generater   r   r   r   r   r   r	   r
   r   r   r   r   	primetestr   r   r   factor_r   r   r   r   r   r   r   r   r   r   r   r   r   r   r    r!   r"   r#   r$   r%   r&   r'   r(   r)   r*   r+   r,   partitions_r-   residue_ntheoryr.   r/   r0   r1   r2   r3   r4   r5   r6   r7   r8   r9   r:   r;   r<   multinomialr=   r>   r?   rD   r@   rA   rB   rC   rF   rE   rG   rH   rI   rJ   rK   __all__     N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/ntheory/__init__.py<module>rW      s   M M M M D D        %> > > > > 7 7 9 8 0  %rU   