U L?hB@sddlZddlZddlmZdgdZeddD],Zegdde>ede>dded><q0dEddZdFdd Z d d Z d d Z ddZ ddZ ejdddkrejZejZnddlmZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd'd(Zd)d*Zd+d,Zd-d.Zd/d0Zd1d2Zd3d4Z d5d6Z!d7d8Z"d9d:Z#d;d<Z$d=d>Z%d?d@Z&dAdBZ'dCdDZ(dS)GNcCs|sdSt||?}|d@}|r,t||Sd|}|dL}|d}|d|>kr\||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_bytetzprH/var/www/html/venv/lib/python3.8/site-packages/sympy/external/ntheory.py bit_scan1s,       rcCst|d|>|S)Nr)r)r r rrr bit_scan01srcCs|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_frrrremove5s0   r!cCsttt|S)z Return x!.)intmlibZifacr rrr factorialQsr%cCsttt|S)zInteger square root of x.)r"r#isqrtr$rrrsqrtVsr'cCs"tt|\}}t|t|fS)z'Integer square root of x and remainder.r#sqrtremr")r srrrrr)[sr)r) reducecGsttj|dS)zgcd of multiple integers.r)r/mathgcdargsrrrr1ksr1cGsd|kr dStdd|dS)zlcm of multiple integers.rcSs||t||S)N)r0r1r rrrrtzlcm..rr.r2rrrlcmpsr7cCs|dkrd| fSd|fS)Nrrrrr rrr_signws r9c Cs|r|s2t|pt|}|s dS|||||fSt|\}}t|\}}d\}}d\}}|rt||\} } || }}||| |}}||| |}}qZ|||||fS)N)rrr)rrrr)rr9r) argZx_signZy_signr r+rr*qcrrrgcdext}s   r?cCsz|dkr dSdd|d@>@r dS|d}dd|d>@r@rPdSd d|d >@rddStt|ddkS) z$Return True if x is a square number.rFl }{wo^?{~riE l}}k-[o{?_}cl=}:Mv?_[l}s;yUr(r rrrr is_squaresrEcCs0zt|d|WStk r*tdYnXdS)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)powrZeroDivisionErrorrDrrrinvertsrHcCsH|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)rrFr4rrrlegendresrIcCs|dks|dstd||;}|s0t|dkS|dks@|dkrDdSt||dkrVdSd}|dkr|ddkr|dkr|dL}|ddkrb| }qb||}}|d|dkrdkrnn| }||;}qZ|S) zJacobi symbol (x / y).rrz#y should be an odd positive integerrrr,rr,)rr"r1)r rjrrrjacobis(     rMcCsvt||dkrdS|dkrdS|dkr2|dkr2dnd}t|}t|}||L}|drh|ddkrh| }|t||S)zKronecker symbol (x / y).rrrrrrJ)r1rrrM)r rsignr*rrr kroneckersrOc Cs|dkrtd|dkr td|dkr0|dfS|dkr@|dfS|dkrdt|\}}t|| fS||krtdSzt|d |d }Wn\tk rt||}|d krt|d }td ||d|>}n td |}YnX|d krDd|}}||d}||d||||}}t||dkrqHqn|}||}||krn|d7}||}qP||kr|d8}||}qn|||kfS)Nrzy must be nonnegativerzn must be positiver:Tr)rFg?g?5g@l r) rr#r)r"r OverflowErrorr0log2r) rr r rguessexpshiftZxprevr rrrirootsH          rVcCsr|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)rr1rFr r;rrr is_fermat_prp)s rXcCs||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)rr1rFrMrWrrr is_euler_prp8s rYcCsvt|d}t|||?|}|dks0||dkr4dSt|dD]0}t|d|}||dkrbdS|dkr@dSq@dS)NrTrF)rrFrange)r r;r*_rrr_is_strong_prpGs   r\cCsh|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)rr1r\rWrrr is_strong_prpUs r]c 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)rrrrrKrr,N1r)bin) r PQkDUVQkrrrr_lucas_sequencedsv9              rgcCsz|dd|}|dks(|dks(|dkr0td|dkr@td|dkrLdS|ddkr`|dkSt||||d||kS) NrrKr)rrz,invalid values for p,q in is_fibonacci_prp()rz1is_fibonacci_prp() requires 'n' be greater than 0F)rrgr rr=drrris_fibonacci_prps rjcCs|dd|}|dkr td|dkr0td|dkrsN  *  $. (