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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 exists)powrZeroDivisionErrorrJrrrinverts  rNcCsH|dks|ds td||;}|sdSt||dd|dkr"dSdS)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)rrLr9rrrlegendresrOcCs|dks|ds td||;}|st|dkS|dks |dkr"dSt||dkr+dSd}|dkrr|ddkrR|dkrR|dL}|ddvrH| }|ddkrR|dks;||}}|d|dkredkrjnn| }||;}|dks1|S) zJacobi symbol (x / y).rrz#y should be an odd positive integerrrr1rr1)rr%r6)r rjrrrjacobis,     rScCsvt||dkr dS|dkrdS|dkr|dkrdnd}t|}t|}||L}|dr4|ddvr4| }|t||S)zKronecker symbol (x / y).rrrrrrP)r6rrrS)r rsignr/rrr kroneckersrUc Cs|dkrtd|dkrtd|dvr|dfS|dkr |dfS|dkr2t|\}}t|| fS||kr:dSz t|d |d }Wn-tyst||}|d krkt|d }td ||d|>}ntd |}Ynw|d krd|}} ||d}||d||||}}t||dkrnq~n|}||}||kr|d7}||}||ks||kr|d8}||}||ks|||kfS)Nrzy must be nonnegativerzn must be positiver>Tr)rFg?g?5g@l r) rr&r.r%r OverflowErrorr5log2r) rr r r guessexpshiftxprevr rrrirootsV       r]cCsr|dkrtd|dkrtd|dkrdS|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)rr6rLr r?rrr is_fermat_prp)s r_cCs||dkrtd|dkrtd|dkrdS|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)rr6rLrSr^rrr is_euler_prp8s r`cCsvt|d}t|||?|}|dks||dkrdSt|dD]}t|d|}||dkr1dS|dkr8dSq dS)NrTrF)rrLrange)r r?r/_rrr_is_strong_prpGs   rccCsh|dkrtd|dkrtd|dkrdS|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)rr6rcr^rrr is_strong_prpUs  rdc Cs|dkrdS|dd|}d}|}||}|dkr`t|ddD]<}|||}||d|}|dkr^|||||||}}|d@rM||7}|d@rU||7}|d?|d?}}q"n|dkr|d krt|ddD]>}|||}|dkr||d|}n ||d|}d}|dkr|||d>}}|d@r||7}|dL}||7}d }qp||;}n|dkrt|ddD]=}|||}||d||}||9}|dkr||||d>}}|d@r||7}|dL}||}||9}||;}qnWt|ddD]N}|||}||d||}||9}|dkrP|||||||}}|d@r:||7}|d@rC||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)rrrrrQrr1N1r)bin) r PQkDUVQkrrrr_lucas_sequenceds~9           rncCsz|dd|}|dks|dks|dvrtd|dkr td|dkr&dS|ddkr0|dkSt||||d||kS) NrrQr)rrz,invalid values for p,q in is_fibonacci_prp()rz1is_fibonacci_prp() requires 'n' be greater than 0F)rrnr rrCdrrris_fibonacci_prps rqcCs|dd|}|dkrtd|dkrtd|dkrdS|ddkr(|dkSt|||d|fvr7tdt||||t||ddkS) NrrQrz(invalid values for p,q in is_lucas_prp()rz-is_lucas_prp() requires 'n' be greater than 0Fz)is_lucas_prp() requires gcd(n,2*q*D) == 1)rr6rnrSrorrr is_lucas_prps  rrcCstdddD];}|d@r| }t||}|dkr+t|dd|d|dddkS|dkr6||r6dS|d krAt|rAdSqtd ) adLucas compositeness test with the Selfridge parameters for n. Explanation =========== The Lucas compositeness test checks whether n is a prime number. The test can be run with arbitrary parameters ``P`` and ``Q``, which also change the performance of the test. So, which parameters are most effective for running the Lucas compositeness test? As an algorithm for determining ``P`` and ``Q``, Selfridge proposed method A [1]_ page 1401 (Since two methods were proposed, referred to simply as A and B in the paper, we will refer to one of them as "method A"). method A fixes ``P = 1``. Then, ``D`` defined by ``D = P**2 - 4Q`` is varied from 5, -7, 9, -11, 13, and so on, with the first ``D`` being ``jacobi(D, n) == -1``. Once ``D`` is determined, ``Q`` is determined to be ``(P**2 - D)//4``. References ========== .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes, Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417, https://doi.org/10.1090%2FS0025-5718-1980-0583518-6 http://mpqs.free.fr/LucasPseudoprimes.pdf r@BrrrrQrF z=appropriate value for D cannot be found in is_selfridge_prp())rarSrnrKr)r rjrRrrr_is_selfridge_prps &rucC8|dkrtd|dkrdS|ddkr|dkSt|S)Nr1is_selfridge_prp() requires 'n' be greater than 0Frr)rrur<rrris_selfridge_prp' rxc Cs|dd|}|dkrtd|dkrtd|dkrdS|ddkr(|dkSt|||d|fvr7tdt||}t||}t||||||?\}}}|dksX|dkrZd St|dD]} ||d||}|dkrsd St|d|}q`dS) NrrQrz/invalid values for p,q in is_strong_lucas_prp()rrwFz0is_strong_lucas_prp() requires gcd(n,2*q*D) == 1T)rr6rSrrnrarL) r rrCrjrRr/rkrlrmrbrrris_strong_lucas_prp1s,   rzcCstdddD]o}|d@r| }t||}|dkr_t|d}t|dd|d|d|?\}}}|dks8|dkr;dSt|dD]}||d||}|dkrUdSt|d|}qAd S|dkrj||rjd S|d krut|rud Sqtd ) NrrsrrrrQrTFrtzDappropriate value for D cannot be found in is_strong_selfridge_prp())rarSrrnrLrKr)r rjrRr/rkrlrmrbrrr_is_strong_selfridge_prpJs*  $r{cCrv)Nrz8is_strong_selfridge_prp() requires 'n' be greater than 0Frr)rr{r<rrris_strong_selfridge_prpbryr|cCB|dkrtd|dkrdS|ddkr|dkSt|do t|S)Nrz,is_bpsw_prp() requires 'n' be greater than 0Frr)rrcrur<rrr is_bpsw_prpl r~cCr})Nrz3is_strong_bpsw_prp() requires 'n' be greater than 0Frr)rrcr{r<rrris_strong_bpsw_prpvrr)r))sysr5 mpmath.libmplibmpr&rrarRrrr#r)r,r. version_infor6r; functoolsr4r=rErKrNrOrSrUr]r_r`rcrdrnrqrrrurxrzr{r|r~rrrrrsP  *   $. 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