\h@ZSSKrSSKJr S/S-r\"SS5Hr\/SS\- --\S\-SS\S--2'M S"SjrS"SjrSr S r S r S r \Rr \RrS rS rSrSrSrSrSrSrSrSrSrSrSrSrSrSrSrSr Sr!Sr"S r#S!r$g)#NcU(dg[X- 5nUS-nU(a [UU-$SU-nUS-nUR5S- nUSU-:XaXC-$US:a!US-(dUS-nUS- nUS-(dMOJUS- nUS-(d;USU-S- -(aUS-nUSU-S- -(aMX-nX5- nUS-(dM;U[US--$)Nrri,)abs_small_trailing bit_length)xnlow_bytetzps N/var/www/auris/envauris/lib/python3.13/site-packages/sympy/external/ntheory.py bit_scan1rs  AF A4xHx(1,, AA!GA AAF{u 3wd( !GA FAd(( Fd(Q!|$aQ!|$$ GA FA d(( q4x( ((c&[USU--U5$)Nr)r)r r s r bit_scan0r0s Q!q&\1 %%rcUS:a [S5eUS:XagUS:Xa[U5nX- U4$Sn[X5upEU(dUnUS- nUS:agUS-/nU(aZUSn[X5upEU(d(US[U5-- nUnUR US-5 OUR 5 U(aMZ[X5upEU(dMX4$)Nzfactor must be > 1r)rrr) ValueErrorrdivmodlenappendpop)r fbmyrempow_list_fs rremover'4s1u-..AvAv aLvqy A A\FA  Q q51vHb\c(m++AAOOBE*LLN(c 4KrcR[[R"[U555$)z Return x!.)intmlibifacr s r factorialr-Ps tyyQ !!rcR[[R"[U555$)zInteger square root of x.)r)r*isqrtr,s rsqrtr0Us tzz#a&! ""rcp[R"[U55up[U5[U54$)z'Integer square root of x and remainder.r*sqrtremr))r srs rr3r3Zs) <<A DA FCF rc US:aSU*4$SU4$)Nrrrr s r_signr9ds1uA2v a4Krc*U(aU(d.[U5=(d [U5nU(dgX U-X-4$[U5up0[U5upASupVSupxU(a&[X5upXpXeX-- peXX-- pU(aM&XU-Xt-4$)N)rrr)rrrr)r r9r) ar!gx_signy_signr r5r#r4qcs rgcdextrBjs A F c!f616""aIFaIF DA DA a|1ac'1ac'1 ! 6z1: &&rcUS:agSSUS---(agUS-nSSUS---(agS SUS ---(agS SUS ---(ag[R"[U55SS:H$) z$Return True if x is a square number.rFl }{wo^?{~riE l}}k-[o{?_}cl=}:Mv?_[l}s;yUr2r r"s r is_squarerIs1u**Q1s7^< F A"aAFm4 A!b&M2!B-0 <<A  "a ''rcP[USU5$![a [S5ef=f)zModular inverse of x modulo m. Returns y such that x*y == 1 mod m. Uses ``math.pow`` but reproduces the behaviour of ``gmpy2.invert`` which raises ZeroDivisionError if no inverse exists. rzinvert() no inverse exists)powrZeroDivisionErrorrHs rinvertrMs0>1b!} > <==>s %cUS::d US-(d [S5eX-nU(dg[XS- S-U5S:Xagg)ztLegendre symbol (x / y). Following the implementation of gmpy2, the error is raised only when y is an even number. rrzy should be an odd primerr)rrK)r r#s rlegendrerOsG  AvQU344FA  11ulA!# rchUS::d US-(d [S5eX-nU(d[US:H5$US:XdUS:Xag[X5S:wagSnUS:waXUS-S:Xa(US:a"US-nUS-S;aU*nUS-S:XaUS:aM"XpUS-US-s=:XaS:XaO OU*nX-nUS:waMXU$) zJacobi symbol (x / y).rrz#y should be an odd positive integerrrrrR)rr)gcd)r r#js rjacobirVsAvQU>??FA 16{Ava 1yA~ A q&!eqjQU !GA1uB!eqjQU1 q5AE Q A  q& Hrc[X5S:wagUS:XagUS:aUS:aSOSn[U5n[U5nX-nUS-(a US-S;aU*nU[X5-$)zKronecker symbol (x / y).rrrrrrQ)rTr rrV)r r#signr4s r kroneckerrYsm 1yA~AvQ1q52aD AA! AGA1uQ&u &, rcUS:a [S5eUS:a [S5eUS;aUS4$US:XaUS4$US:Xa*[R"U5up#[U5U(+4$XR 5:ag[US U- -S -5nUS :a/SUp'X!S- -nX!S- U-X--U-p'[X'- 5S:aOM+UnX!-nX:aUS- nX!-nX:aMX:aUS-nX!-nX:aMX(U:H4$![ aV [ R"U5U- nUS :a&[US - 5n[S XV- -S-5U-nN[S U-5nNf=f)Nrzy must be nonnegativerzn must be positiver;Tr)rFg?g?5g@l r) rr*r3r)r OverflowErrormathlog2r ) r#r r r$guessexpshiftxprevrs rirootrcs1u0111u-..F{$wAv$wAva1v3wLLN"A1IO$ u}uqE AE19qt+a/119~!   A % Q D % % Q D % 1f93 "iil1n 8bMEck*Q./58ESME "s<C;;A E EEcUS:a [S5eUS:a [S5eUS:XagUS-S:XaUS:H$X-n[X5S:wa [S5e[XS- U5S:H$)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)rrTrKr r<s r is_fermat_prprfsz1uRSS1uIJJAv1uzAv FA 1yA~ABB qa% q  rcUS:a [S5eUS:a [S5eUS:XagUS-S:XaUS:H$X-n[X5S:wa [S5e[XS- U5[X5U-:H$)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)rrTrKrVres r is_euler_prprh%s1uQRR1uHIIAv1uzAv FA 1yA~@AA qq&! q q 0 00rc[US- 5n[XU- U5nUS:XdXS- :Xag[US- 5H"n[USU5nXS- :Xa gUS:XdM" g g)NrTrF)rrKrange)r r<r4_s r_is_strong_prprl4sl!a%A AAvqAAv!e 1q5\ 1aL A: 6  rcUS:a [S5eUS:a [S5eUS:XagUS-S:XaUS:H$X-n[X5S:wa [S5e[X5$)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)rrTrlres r is_strong_prprnBso1uRSS1uIJJAv1uzAv FA 1yA~ABB ! rcUS:XagUS-SU-- nSnUnX -nUS:Xad[U5SSHPnXV-U-nXf-S- U-nUS:XdMXQ-U-Xa-XT--peUS-(aXP- nUS-(aX`- nUS- US- peMR GOEUS:XamUS :Xag[U5SSHPnXV-U-nUS:Xa Xf-S- U-nO Xf-S-U-nSnUS:XdM/XV-US-peUS-(aXP- nUS-nXe- nS nMR Xp-nOUS:Xa][U5SSHJnXV-U-nXf-SU-- U-nXw-nUS:Xa%XV-X%-S-peUS-(aXP- nUS-nXV- nXr-nXp-nML Oo[U5SSH]nXV-U-nXf-SU-- U-nXw-nUS:Xa8XQ-U-Xa-XT--peUS-(aXP- nUS-(aX`- nUS- US- peXr-nXp-nM_ XP-X`-U4$) aReturn the modular Lucas sequence (U_k, V_k, Q_k). Explanation =========== Given a Lucas sequence defined by P, Q, returns the kth values for U and V, along with Q^k, all modulo n. This is intended for use with possibly very large values of n and k, where the combinatorial functions would be completely unusable. .. math :: U_k = \begin{cases} 0 & \text{if } k = 0\\ 1 & \text{if } k = 1\\ PU_{k-1} - QU_{k-2} & \text{if } k > 1 \end{cases}\\ V_k = \begin{cases} 2 & \text{if } k = 0\\ P & \text{if } k = 1\\ PV_{k-1} - QV_{k-2} & \text{if } k > 1 \end{cases} The modular Lucas sequences are used in numerous places in number theory, especially in the Lucas compositeness tests and the various n + 1 proofs. Parameters ========== n : int n is an odd number greater than or equal to 3 P : int Q : int D determined by D = P**2 - 4*Q is non-zero k : int k is a nonnegative integer Returns ======= U, V, Qk : (int, int, int) `(U_k \bmod{n}, V_k \bmod{n}, Q^k \bmod{n})` Examples ======== >>> from sympy.external.ntheory import _lucas_sequence >>> N = 10**2000 + 4561 >>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol (0, 2, 1) References ========== .. [1] https://en.wikipedia.org/wiki/Lucas_sequence r)rrrrrSrrRN1r)bin) r PQkDUVQkr!s r_lucas_sequenceryQspr Av 1qs A A A BAvQA AqA ACxsQwac 1q5FAq5FAAvqAv1 aAGQA AQwS1WMS1WMCxqAv1q5FAa  aQA Aqtq A HBCxz1q5FAaE GB QA Aqtq A HBCxsQwac 1q5FAq5FAAvqAv1 GB E15" rcUS-SU-- nUS:Xd US::dUS;a [S5eUS:a [S5eUS:XagUS-S:XaUS:H$[XX 5SX-:H$) NrrSr)rrz,invalid values for p,q in is_fibonacci_prp()rz1is_fibonacci_prp() requires 'n' be greater than 0F)rryr rr@ds ris_fibonacci_prpr}s} 1qs AAva1G+GHH1uLMMAv1uzAv 1 &q )QU 22rc US-SU-- nUS:Xa [S5eUS:a [S5eUS:XagUS-S:XaUS:H$[XU-5SU4;a [S5e[XX [X05- 5SS:H$) NrrSrz(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)rrTryrVr{s r is_lucas_prprs 1qs AAvCDD1uHIIAv1uzAv  1c{1a& DEE 1q $4 5a 8A ==rc[SSS5HknUS-(aU*n[X5nUS:Xa[USSU- S-US-5SS:Hs $US:Xa X-(a gUS :XdMY[U5(dMk g [ S 5e) aLucas compositeness test with the Selfridge parameters for n. Explanation =========== The Lucas compositeness test checks whether n is a prime number. The test can be run with arbitrary parameters ``P`` and ``Q``, which also change the performance of the test. So, which parameters are most effective for running the Lucas compositeness test? As an algorithm for determining ``P`` and ``Q``, Selfridge proposed method A [1]_ page 1401 (Since two methods were proposed, referred to simply as A and B in the paper, we will refer to one of them as "method A"). method A fixes ``P = 1``. Then, ``D`` defined by ``D = P**2 - 4Q`` is varied from 5, -7, 9, -11, 13, and so on, with the first ``D`` being ``jacobi(D, n) == -1``. Once ``D`` is determined, ``Q`` is determined to be ``(P**2 - D)//4``. References ========== .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes, Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417, https://doi.org/10.1090%2FS0025-5718-1980-0583518-6 http://mpqs.free.fr/LucasPseudoprimes.pdf r@BrrrrSrF z=appropriate value for D cannot be found in is_selfridge_prp())rjrVryrIr)r rurUs r_is_selfridge_prprs41i # q5A 1L 7"1a!A#!QU;A>!C C 6ae 7y||$ T UUrcdUS:a [S5eUS:XagUS-S:XaUS:H$[U5$)Nr1is_selfridge_prp() requires 'n' be greater than 0Frr)rrr8s ris_selfridge_prprs>1uLMMAv1uzAv Q rcUS-SU-- nUS:Xa [S5eUS:a [S5eUS:XagUS-S:XaUS:H$[XU-5SU4;a [S5e[X05n[X- 5n[ XX U- U- 5upgnUS:XdUS:Xag [ US- 5H%n Xw-SU-- U-nUS:Xa g [ USU5nM' g) NrrSrz/invalid values for p,q in is_strong_lucas_prp()rrFz0is_strong_lucas_prp() requires gcd(n,2*q*D) == 1T)rrTrVrryrjrK) r rr@rurUr4rvrwrxrks ris_strong_lucas_prprs 1qs AAvJKK1uLMMAv1uzAv  1c{1a& KLLq A!%AqQQ1 5HA"Ava 1q5\ S1R4Z1  6 Q]  rc[SSS5HnUS-(aU*n[X5nUS:Xas[US-5n[USSU- S-US-U- 5upEnUS:XdUS:Xa g[US- 5H&nXU-SU-- U-nUS:Xa g[ USU5nM( g US:Xa X-(a g US :XdM[ U5(dM g [ S 5e) NrrrrrrSrTFrzDappropriate value for D cannot be found in is_strong_selfridge_prp())rjrVrryrKrIr)r rurUr4rvrwrxrks r_is_strong_selfridge_prpr7s 1i # q5A 1L 7!a% A&q!acaZ!a%AFHA"Ava1q5\S1R4Z1$6Q] "  6ae 7y||'$( [ \\rcdUS:a [S5eUS:XagUS-S:XaUS:H$[U5$)Nrz8is_strong_selfridge_prp() requires 'n' be greater than 0Frr)rrr8s ris_strong_selfridge_prprOs>1uSTTAv1uzAv #A &&rcUS:a [S5eUS:XagUS-S:XaUS:H$[US5=(a [U5$)Nrz,is_bpsw_prp() requires 'n' be greater than 0Frr)rrlrr8s r is_bpsw_prprYsK1uGHHAv1uzAv !Q  8$5a$88rcUS:a [S5eUS:XagUS-S:XaUS:H$[US5=(a [U5$)Nrz3is_strong_bpsw_prp() requires 'n' be greater than 0Frr)rrlrr8s ris_strong_bpsw_prprcsK1uNOOAv1uzAv !Q  ?$rs  #) q!A/0cQ1q5\.BOAF*aAEl*+ )@&8" #   hh hh '*!(H >   0 +\ ! 1   |~ 3 >%VP 2]0'9@r