gBddlZddlZddlmZdgdzZeddD]Zegddez zzedezddedzz<d'dZd'dZ dZ d Z d Z d Z ejdd d krejZejZn ddlmZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZdZ dZ!d Z"d!Z#d"Z$d#Z%d$Z&d%Z'd&Z(dS)(Nc~|sdSt||z }|dz}|rt||zSd|z}|dz}|dz }|d|zkr||zS|dkr|dzs|dz}|dz }|dzn4|dz }|dzs*|d|zdz zr|dz}|d|zdz z||z}||z }|dz*|t|dzzS)Nrri,)abs_small_trailing bit_length)xnlow_bytetzps R/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/external/ntheory.py bit_scan1rs1  AF A4xH-x(1,, AA!GA AAF{{1u 3wwd(  !GA FAd(  Fd( Q!|$ aQ!|$  !GA FA d(  q4x( ((c.t|d|zz|S)Nr)r)r r s r bit_scan0r1s Q!q&\1 % %%rc|dkrtd|dkrdS|dkrt|}||z |fSd}t||\}}|s|}|dz }|dkrk|dzg}|rc|d}t||\}}|s0|dt|zz }|}||dzn||ct||\}}|||fS)Nzfactor must be > 1r)rrr) ValueErrorrdivmodlenappendpop)r fbmyrempow_list_fs rremover'5s&1uu-...AvvtAvv aLLAvqy A Aq\\FAs  Q q551vH #b\23#c(mm++AAOOBE****LLNNN #13 a4Krc^ttjt|S)z Return x!.)intmlibifacr s r factorialr-Qs tyQ  ! !!rc^ttjt|S)zInteger square root of x.)r)r*isqrtr,s rsqrtr0Vs tz#a&&!! " ""rctjt|\}}t|t|fS)z'Integer square root of x and remainder.r*sqrtremr))r srs rr3r3[s2 <A  DAq FFCFF rr) reducec8ttj|dS)zgcd of multiple integers.r)r9mathgcdargss rr<r<ksdha(((rc2d|vrdStd|dS)zlcm of multiple integers.rc8||ztj||zS)N)r;r<r r#s rzlcm..ts1Q3A#6rrr8r=s rlcmrCps& 99166a@@@rc |dkrd| fSd|fS)Nrrrr s r_signrGws1uuA2v a4Krc4|r|s-t|pt|}|sdS|||z||zfSt|\}}t|\}}d\}}d\}}|r-t||\} } || }}||| |zz }}||| |zz }}|-|||z||zfS)N)rrr)rrrr)r rGr) ar!gx_signy_signr r5r#r4qcs rgcdextrP}s #A# FF c!ff 91616""aIFAaIFA DAq DAq a||1!1!ac'1!ac'1  q6z1v: &&rc|dkrdSdd|dzzzrdS|dz}dd|dzzzrdSd d|d zzzrdSd d|d zzzrdStjt|ddkS) z$Return True if x is a square number.rFl }{wo^?{~riE l}}k-[o{?_}cl=}:Mv?_[l}s;yUr2r r"s r is_squarerWs1uuu**Q1s7^<u F A"aAFm4u A!b&M2u!B-0u <A   "a ''rc` t|d|S#t$rtdwxYw)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 exists)powrZeroDivisionErrorrVs rinvertr[sA>1b!}} >>> <===>s-c|dks|dzstd||z}|sdSt||dz dz|dkrdSdS)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)rrYrAs rlegendrer]sc  AvvQUv3444FA q 1q1ulA!##q 2rcp|dks|dzstd||z}|st|dkS|dks|dkrdSt||dkrdSd}|dkrU|dzdkr$|dkr|dz}|dzdvr| }|dzdkr|dk||}}|dz|dzcxkrdkrnn| }||z}|dkU|S) zJacobi symbol (x / y).rrz#y should be an odd positive integerrrr6rr6)rr)r<)r r#js rjacobirbsAvvQUv>???FA 16{{Avvaq 1ayyA~~q A q&&!eqjjQUU !GA1uB!eqjjQUU!1 q5AE    Q     A Q q&& Hrct||dkrdS|dkrdS|dkr|dkrdnd}t|}t|}||z}|dzr |dzdvr| }|t||zS)zKronecker symbol (x / y).rrrrrr_)r<r rrb)r r#signr4s r kroneckerres 1ayyA~~qAvvqQ1q5522aD AA! A!GA1uQ&u &A,, rc|dkrtd|dkrtd|dvr|dfS|dkr|dfS|dkr)tj|\}}t|| fS||krdS t|d |z zd z}nm#t $r`t j||z }|d kr.t|d z }td ||z zdz|z}ntd |z}YnwxYw|d kr9d|}} ||dz z}||dz |z||zz|z}}t||z dkrn3n|}||z}||kr|dz }||z}||k||kr|dz}||z}||k|||kfS)Nrzy must be nonnegativerzn must be positiverITr)rFg?g?5g@l r) rr*r3r)r OverflowErrorr;log2r ) r#r r r$guessexpshiftxprevrs rirootrns1uu01111uu-...F{{$wAvv$wAvva31vv3wALLNNx"A1IO$$ """ill1n 88bMMEcEk*Q.//58EESMME " u}}uq AE AAE19q!t+a/1E1u9~~!!    1A a%% Q qD a%% a%% Q qD a%% a1f9sB A'D  D c|dkrtd|dkrtd|dkrdS|dzdkr|dkS||z}t||dkrtdt||dz |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)rr<rYr rJs r is_fermat_prprq)s1uuRSSS1uuIJJJAvvu1uzzAv FA 1ayyA~~ABBB q!a%  q  rc&|dkrtd|dkrtd|dkrdS|dzdkr|dkS||z}t||dkrtdt||dz |t|||zkS)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)rr<rYrbrps r is_euler_prprs8s1uuQRRR1uuHIIIAvvu1uzzAv FA 1ayyA~~@AAA q!q&!  q! q 0 00rct|dz }t|||z |}|dks ||dz krdSt|dz D](}t|d|}||dz krdS|dkrdS)dS)NrTrF)rrYrange)r rJr4_s r_is_strong_prprwGs!a%A AqAvqAAvva!et 1q5\\ 1aLL A::44 6655  5rc|dkrtd|dkrtd|dkrdS|dzdkr|dkS||z}t||dkrtdt||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)rr<rwrps r is_strong_prpryUs1uuRSSS1uuIJJJAvvu1uzzAv FA 1ayyA~~ABBB !Q  rc|dkrdS|dzd|zz }d}|}||z}|dkrft|ddD]L}||z|z}||zdz |z}|dkr1||z|z||z||zz}}|dzr||z }|dzr||z }|dz |dz }}Mn\|dkrs|d krmt|ddD]O}||z|z}|dkr ||zdz |z}n ||zdz|z}d}|dkr ||z|dz}}|dzr||z }|dz}||z }d }P||z}n|dkrgt|ddD]N}||z|z}||zd|zz |z}||z}|dkr&||z||zdz}}|dzr||z }|dz}||z }||z}||z}Onvt|ddD]^}||z|z}||zd|zz |z}||z}|dkr6||z|z||z||zz}}|dzr||z }|dzr||z }|dz |dz }}||z}||z}_||z||z|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`rr6N1r)bin) r PQkDUVQkr!s r_lucas_sequencerds,r Avvy 1qs A A A QBAvvQ & &A1 A1qA ACxxsQw!ac 1q5FAq5FAAvqAv1 & aAGGQ  A1 AQwwqS1WMqS1WMCxxAqAv1q5FAaQ a aQ  A1 A1qtq A "HBCxxA!z1q5FAaEa !GBB  Q  A1 A1qtq A "HBCxxsQw!ac 1q5FAq5FAAvqAv1a !GBB E1q5" rc|dzd|zz }|dks |dks|dvrtd|dkrtd|dkrdS|dzdkr|dkSt||||d||zkS) Nrr`r)rrz,invalid values for p,q in is_fibonacci_prp()rz1is_fibonacci_prp() requires 'n' be greater than 0F)rrr rrNds ris_fibonacci_prprs 1qs AAvva1G++GHHH1uuLMMMAvvu1uzzAv 1aA & &q )QU 22rc @|dzd|zz }|dkrtd|dkrtd|dkrdS|dzdkr|dkSt|||zd|fvrtdt||||t||z ddkS) Nrr`rz(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)rr<rrbrs r is_lucas_prprs 1qs AAvvCDDD1uuHIIIAvvu1uzzAv  1ac{{1a&  DEEE 1aAq! $4 5 5a 8A ==rctdddD]m}|dzr| }t||}|dkr't|dd|z dz|dzddkcS|dkr||zrdS|d krt|rdSnt d ) 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@Brrrr`rF z=appropriate value for D cannot be found in is_selfridge_prp())rurbrrWr)r rras r_is_selfridge_prprs41i # #   q5 A 1aLL 77"1a!A#!QU;;A>!C C C C 66a!e655 77y||755 T U UUrcx|dkrtd|dkrdS|dzdkr|dkSt|S)Nr1is_selfridge_prp() requires 'n' be greater than 0Frr)rrrFs ris_selfridge_prpr'sL1uuLMMMAvvu1uzzAv Q  rc|dzd|zz }|dkrtd|dkrtd|dkrdS|dzdkr|dkSt|||zd|fvrtdt||}t||z }t |||||z |z \}}}|dks|dkrd St |dz D]*} ||zd|zz |z}|dkrd St |d|}+dS) Nrr`rz/invalid values for p,q in is_strong_lucas_prp()rrFz0is_strong_lucas_prp() requires gcd(n,2*q*D) == 1T)rr<rbrrrurY) r rrNrrar4rrrrvs ris_strong_lucas_prpr1s; 1qs AAvvJKKK1uuLMMMAvvu1uzzAv  1ac{{1a&  KLLLq! A!a%Aq!QQ1 55HAq"Avvat 1q5\\ qS1R4Z1  6644 Q]] 5rctdddD]}|dzr| }t||}|dkrt|dz}t|dd|z dz|dz|z \}}}|dks|dkrdSt|dz D]+}||zd|zz |z}|dkrdSt |d|},d S|dkr||zrd S|d krt |rd St d ) Nrrrrrr`rTFrzDappropriate value for D cannot be found in is_strong_selfridge_prp())rurbrrrYrWr)r rrar4rrrrvs r_is_strong_selfridge_prprJs3 1i # # q5 A 1aLL 77!a%  A&q!acaZ!a%AFFHAq"Avvatt1q5\\ # #qS1R4Z1$66444Q]]55 66a!e655 77y||755 [ \ \\rcx|dkrtd|dkrdS|dzdkr|dkSt|S)Nrz8is_strong_selfridge_prp() requires 'n' be greater than 0Frr)rrrFs ris_strong_selfridge_prprbsL1uuSTTTAvvu1uzzAv #A & &&rc|dkrtd|dkrdS|dzdkr|dkSt|dot|S)Nrz,is_bpsw_prp() requires 'n' be greater than 0Frr)rrwrrFs r is_bpsw_prprls\1uuGHHHAvvu1uzzAv !Q   8$5a$8$88rc|dkrtd|dkrdS|dzdkr|dkSt|dot|S)Nrz3is_strong_bpsw_prp() requires 'n' be greater than 0Frr)rrwrrFs ris_strong_bpsw_prprvs\1uuNOOOAvvu1uzzAv !Q   ?$rs  #) q!CCA/0cQ1q5\.BOAF*aAEl*++))))@&&&&8""" ###  BQB6!! 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