a kh@@sJddlZddlmZdgdZeddD],Zegdde>ede>dded><q(d>ddZd?dd Zd d Z d d Z ddZ ddZ ej Z ejZddZddZddZddZddZddZddZd d!Zd"d#Zd$d%Zd&d'Zd(d)Zd*d+Zd,d-Zd.d/Zd0d1Zd2d3Zd4d5Z d6d7Z!d8d9Z"d:d;Z#dkr\||S|dkr|d@s|dL}|d7}qdn<|d?}|d@s|d|>d@r|dL}q||L}||7}q|t|d@S)Nrri,)abs_small_trailing bit_length)xnZlow_bytetzprD/var/www/auris/lib/python3.9/site-packages/sympy/external/ntheory.py bit_scan1s,       rcCst|d|>|S)Nr)r)r r rrr bit_scan00srcCs|dkrtd|dkrdS|dkr8t|}||?|fSd}t||\}}|s|}|d7}|dkr|dg}|r|d}t||\}}|s|dt|>7}|}||dql|qlt||\}}qJ||fS)Nzfactor must be > 1r)rrr) ValueErrorrdivmodlenappendpop)r fbmyremZpow_list_frrrremove4s0   r!cCsttt|S)z Return x!.)intmlibZifacr rrr factorialPsr%cCsttt|S)zInteger square root of x.)r"r#isqrtr$rrrsqrtUsr'cCs"tt|\}}t|t|fS)z'Integer square root of x and remainder.r#sqrtremr")r srrrrr)Zsr)cCs|dkrd| fSd|fS)Nrrrrr rrr_signds r-c Cs|r|s2t|pt|}|s dS|||||fSt|\}}t|\}}d\}}d\}}|rt||\} } || }}||| |}}||| |}}qZ|||||fS)N)rrr)rrrr)rr-r) argZx_signZy_signr r+rr*qcrrrgcdextjs   r3cCsz|dkr dSdd|d@>@r dS|d}dd|d>@r@rPdSd d|d >@rddStt|ddkS) z$Return True if x is a square number.rFl }{wo^?{~riE l}}k-[o{?_}cl=}:Mv?_[l}s;yUr(r rrrr is_squaresr9cCs.zt|d|WSty(tdYn0dS)zModular inverse of x modulo m. Returns y such that x*y == 1 mod m. Uses ``math.pow`` but reproduces the behaviour of ``gmpy2.invert`` which raises ZeroDivisionError if no inverse exists. rzinvert() no inverse existsN)powrZeroDivisionErrorr8rrrinverts r<cCsH|dks|dstd||;}|s(dSt||dd|dkrDdSdS)zLegendre symbol (x / y). Following the implementation of gmpy2, the error is raised only when y is an even number. rrzy should be an odd primerr)rr:)r rrrrlegendresr=cCs|dks|dstd||;}|s0t|dkS|dks@|dkrDdSt||dkrVdSd}|dkr|ddkr|dkr|dL}|ddvrb| }qb||}}|d|dkrdkrnn| }||;}qZ|S) zJacobi symbol (x / y).rrz#y should be an odd positive integerrrrr?)rr"gcd)r rjrrrjacobis(     rCcCsvt||dkrdS|dkrdS|dkr2|dkr2dnd}t|}t|}||L}|drh|ddvrh| }|t||S)zKronecker symbol (x / y).rrrrrr>)rArrrC)r rsignr*rrr kroneckersrEc Cs|dkrtd|dkr td|dvr0|dfS|dkr@|dfS|dkrdt|\}}t|| fS||krtdSzt|d |d }WnZtyt||}|d krt|d }td ||d|>}n td |}Yn0|d kr>d|}}||d}||d||||}}t||dkrqBqn|}||}||krh|d7}||}qJ||kr|d8}||}qh|||kfS)Nrzy must be nonnegativerzn must be positiver.Tr)rFg?g?5g@l r) rr#r)r"r OverflowErrormathlog2r) rr r rguessexpshiftZxprevr rrrirootsH          rMcCsr|dkrtd|dkr td|dkr,dS|ddkr@|dkS||;}t||dkr^tdt||d|dkS)Nrz7is_fermat_prp() requires 'a' greater than or equal to 2rz.is_fermat_prp() requires 'n' be greater than 0Frz&is_fermat_prp() requires gcd(n,a) == 1)rrAr:r r/rrr is_fermat_prps rOcCs||dkrtd|dkr td|dkr,dS|ddkr@|dkS||;}t||dkr^tdt||d?|t|||kS)Nrz6is_euler_prp() requires 'a' greater than or equal to 2rz-is_euler_prp() requires 'n' be greater than 0Frz%is_euler_prp() requires gcd(n,a) == 1)rrAr:rCrNrrr is_euler_prp%s rPcCsvt|d}t|||?|}|dks0||dkr4dSt|dD]0}t|d|}||dkrbdS|dkr@dSq@dS)NrTrF)rr:range)r r/r*_rrr_is_strong_prp4s   rScCsh|dkrtd|dkr td|dkr,dS|ddkr@|dkS||;}t||dkr^tdt||S)Nrz7is_strong_prp() requires 'a' greater than or equal to 2rz.is_strong_prp() requires 'n' be greater than 0Frz&is_strong_prp() requires gcd(n,a) == 1)rrArSrNrrr is_strong_prpBs rTc Cs|dkr dS|dd|}d}|}||}|dkrt|ddD]x}|||}||d|}|dkrD|||||||}}|d@r||7}|d@r||7}|d?|d?}}qDn|dkrt|d krtt|ddD]}|||}|dkr||d|}n||d|}d}|dkr|||d>}}|d@rR||7}|dL}||7}d }q||;}nL|dkrt|ddD]}|||}||d||}||9}|dkr||||d>}}|d@r||7}|dL}||}||9}||;}qnt|ddD]}|||}||d||}||9}|dkr|||||||}}|d@r||7}|d@r||7}|d?|d?}}||9}||;}q"|||||fS) aReturn the modular Lucas sequence (U_k, V_k, Q_k). Explanation =========== Given a Lucas sequence defined by P, Q, returns the kth values for U and V, along with Q^k, all modulo n. This is intended for use with possibly very large values of n and k, where the combinatorial functions would be completely unusable. .. math :: U_k = \begin{cases} 0 & \text{if } k = 0\\ 1 & \text{if } k = 1\\ PU_{k-1} - QU_{k-2} & \text{if } k > 1 \end{cases}\\ V_k = \begin{cases} 2 & \text{if } k = 0\\ P & \text{if } k = 1\\ PV_{k-1} - QV_{k-2} & \text{if } k > 1 \end{cases} The modular Lucas sequences are used in numerous places in number theory, especially in the Lucas compositeness tests and the various n + 1 proofs. Parameters ========== n : int n is an odd number greater than or equal to 3 P : int Q : int D determined by D = P**2 - 4*Q is non-zero k : int k is a nonnegative integer Returns ======= U, V, Qk : (int, int, int) `(U_k \bmod{n}, V_k \bmod{n}, Q^k \bmod{n})` Examples ======== >>> from sympy.external.ntheory import _lucas_sequence >>> N = 10**2000 + 4561 >>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol (0, 2, 1) References ========== .. [1] https://en.wikipedia.org/wiki/Lucas_sequence r)rrrrr@rr?N1r)bin) r PQkDUVQkrrrr_lucas_sequenceQsv9              r^cCsz|dd|}|dks(|dks(|dvr0td|dkr@td|dkrLdS|ddkr`|dkSt||||d||kS) Nrr@r)rrz,invalid values for p,q in is_fibonacci_prp()rz1is_fibonacci_prp() requires 'n' be greater than 0F)rr^r rr1drrris_fibonacci_prps racCs|dd|}|dkr td|dkr0td|dkrsD  * $. (