+ i@^RIt^RIHt^.R,t]!^^4F6t].^^], ,,]^],R^]^,,1&K8 RRltRRltRt Rt Rt Rt ]Pt ]PtR tR tR tR tR tRtRtRtRtRtRtRtRtRtRtRtRtRt Rt!Rt"Rt#Rt$R#) Nc0V'gR#\W, 4pV^,pV'd\V,V,#^V,pV^,pVP4^, pV^V,8Xd WC,#VR8d%V^,'gV^,pV^, pK#MRV^, pV^,'g;V^V,^, ,'d V^,pK(W,pW5, pKIV\V^,,,#)Ni,)abs_small_trailing bit_length)xnlow_bytetzps&& v/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/external/ntheory.py bit_scan1rs  AF A4xHx(1,, AA!GA AAF{u 3wd(( !GA FA Fd((Q!|$$a GA FA q4x( ((c6\V^V,,V4#))r)rr s&&r bit_scan0r0s Q!q&\1 %%rcV^8d \R4hV^8XdR#V^8Xd\V4pW, V3#^p\W4wrEV'gTpV^, pV^8dxV^,.pV'dfVR,p\W4wrEV'g6V^\V4,, pTpVP V^,4K[VP 4Km\W4wrEKW3#)zfactor must be > 1)rr) ValueErrorrdivmodlenappendpop)rfbmyrempow_list_fs&& rremover#4s1u-..Av Av aLvqy A A\FA  Q q51vHb\c(m++AAOOBE*LLN3 4KrcR\\P!\V444#)z Return x!.)intmlibifacrs&r factorialr)Ps tyyQ !!rcR\\P!\V444#)zInteger square root of x.)r%r&isqrtr(s&rsqrtr,Us tzz#a&! ""rcp\P!\V44wr\V4\V43#)z'Integer square root of x and remainder.r&sqrtremr%)rsrs& rr/r/Zs) <<A DA FCF rc"V^8dRV)3#^V3#)rrr s&r_signr5ds1uA2v a4KrcdV'd V'g:\V4;'g \V4pV'gR#W V,W,3#\V4wr0\V4wrA^^re^^rV'd0\W4wrYrYeW,, reYW,, rK7WV,Wt,3#)r)rrr)rr5r) argx_signy_signrr1rr0qcs && rgcdextr=js A F  c!f 616""aIFaIF aq aq a|1ac'1ac'1 6z1: &&rcpV^8dR#R^V^,,,'dR#VR,pR^V^c,,,'dR#R^V^[,,,'dR#R^V^U,,,'dR#\P!\V44^,^8H#)z$Return True if x is a square number.Fl }{wo^?{~iE l}}k-[o{?_}l=}:Mv?_l}s;yr.rrs& r is_squarer@s1u**Q1s7^<< F A"aAFm44 A!b&M22!B-00 <<A  "a ''rcR\VRV4# \d \R4hi;i)zModular inverse of x modulo m. Returns y such that x*y == 1 mod m. Uses ``math.pow`` but reproduces the behaviour of ``gmpy2.invert`` which raises ZeroDivisionError if no inverse exists. zinvert() no inverse existsr)powrZeroDivisionErrorr?s&&rinvertrDs0>1b!} > <==>s &cV^8:gV^,'g \R4hW,pV'g^#\W^, ^,V4^8Xd^#R#)ztLegendre symbol (x / y). Following the implementation of gmpy2, the error is raised only when y is an even number. zy should be an odd primer)rrB)rrs&&rlegendrerFsK  AvQUU344FA  11ulA!# IrcV^8:gV^,'g \R4hW,pV'g\V^8H4#V^8XgV^8Xd^#\W4^8wd^#^pV^8wdcV^,^8Xd&V^8dV^,pV^,R9gK.V)pK3YrV^,V^,u;8Xd^8XdMMV)pW,pKiV#)zJacobi symbol (x / y).z#y should be an odd positive integer)rr%gcd)rrjs&& rjacobirMsAvQUU>??FA 16{Ava 1yA~ A q&!eqjQU !GA1uB1 q5AE Q A  Hrc\W4^8wd^#V^8Xd^#V^8d V^8dRM^p\V4p\V4pW,pV^,'dV^,R9dV)pV\W4,#)zKronecker symbol (x / y).rrH)rKrrrM)rrsignr0s&& r kroneckerrPso 1yA~AvQ1q52aD AA! AGA1uuQ&u &, rc~V^8d \R4hV^8d \R4hVR9dVR3#V^8XdVR3#V^8Xd+\P!V4wr#\V4V'*3#WP 48dR#\VRV, ,R,4pVR 8dQR Tr'W!^, ,pY!^, V,W,,V,r'\W', 4^8gKLMTpW!,pW8dV^, pW!,pKW8dV^,pW!,pKW(V8H3# \ dv\ P!T4T, pT^58d<\T^5, 4p\RYV, ,^,4T,pEL\RT,4pELi;i) rzy must be nonnegativezn must be positiveTg?g?g@)rr)rFl r) rr&r/r%r OverflowErrormathlog2r) rr rr guessexpshiftxprevr s && rirootrYs1u0111u-..F{$wAv$wAva1v3wLLN"A1IO$ u}uqE AE19qt+a/119~!  A % Q D % Q D 1f93 "iil1n 8bMEck*Q./58ESME "s D< 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 :rINN1)rrrr)bin) r PQkDUVQkrs &&&& r_lucas_sequenceroQsr Av 1qs A A A BAvQA AqA ACxsQwac 1q55FAq55FAAvqAv1 aAGQA AQwS1WMS1WMCxqAv1q55FAa  aQA Aqtq A HBCxz1q55FAaE GB QA Aqtq A HBCxsQwac 1q55FAq55FAAvqAv1 GB E15" rcV^,^V,, pV^8XgV^8:gVR9d \R4hV^8d \R4hV^8XdR#V^,^8XdV^8H#\WW 4^,W,8H#)rz,invalid values for p,q in is_fibonacci_prp()z1is_fibonacci_prp() requires 'n' be greater than 0F)rr)rror r r;ds&&& ris_fibonacci_prprss} 1qs AAva1G+GHH1uLMMAv1uzAv 1 &q )QU 22rc DV^,^V,, pV^8Xd \R4hV^8d \R4hV^8XdR#V^,^8XdV^8H#\WV,4^V39d \R4h\WW \W04, 4^,^8H#)rz(invalid values for p,q in is_lucas_prp()z-is_lucas_prp() requires 'n' be greater than 0Fz)is_lucas_prp() requires gcd(n,2*q*D) == 1)rrKrorMrqs&&& r is_lucas_prprus 1qs AAvCDD1uHIIAv1uzAv  1c{1a& DEE 1q $4 5a 8A ==rcP\^R^4FpV^,'dV)p\W4pVR8Xd0\V^^V, ^,V^,4^,^8Hu#V^8XdW,'dR#V^ 8XgKw\V4'gKR# \ R4h)aLucas compositeness test with the Selfridge parameters for n. Explanation =========== The Lucas compositeness test checks whether n is a prime number. The test can be run with arbitrary parameters ``P`` and ``Q``, which also change the performance of the test. So, which parameters are most effective for running the Lucas compositeness test? As an algorithm for determining ``P`` and ``Q``, Selfridge proposed method A [1]_ page 1401 (Since two methods were proposed, referred to simply as A and B in the paper, we will refer to one of them as "method A"). method A fixes ``P = 1``. Then, ``D`` defined by ``D = P**2 - 4Q`` is varied from 5, -7, 9, -11, 13, and so on, with the first ``D`` being ``jacobi(D, n) == -1``. Once ``D`` is determined, ``Q`` is determined to be ``(P**2 - D)//4``. References ========== .. [1] Robert Baillie, Samuel S. Wagstaff, Lucas Pseudoprimes, Math. Comp. Vol 35, Number 152 (1980), pp. 1391-1417, https://doi.org/10.1090%2FS0025-5718-1980-0583518-6 http://mpqs.free.fr/LucasPseudoprimes.pdf @BFz=appropriate value for D cannot be found in is_selfridge_prp()r)r`rMror@r)r rkrLs& r_is_selfridge_prprxs41i # q55A 1L 7"1a!A#!QU;A>!C C 6aee 7y||$ T UUrctV^8d \R4hV^8XdR#V^,^8XdV^8H#\V4#)r1is_selfridge_prp() requires 'n' be greater than 0F)rrxr4s&ris_selfridge_prpr{s>1uLMMAv1uzAv Q rc(V^,^V,, pV^8Xd \R4hV^8d \R4hV^8XdR#V^,^8XdV^8H#\WV,4^V39d \R4h\W04p\W, 4p\ WW V, V, 4wrgpV^8XgV^8XdR#\ V^, 4F7p Ww,^V,, V,pV^8XdR#\ V^V4pK9 R#)rz/invalid values for p,q in is_strong_lucas_prp()rzFz0is_strong_lucas_prp() requires gcd(n,2*q*D) == 1T)rrKrMrror`rB) r r r;rkrLr0rlrmrnras &&& ris_strong_lucas_prpr}s 1qs AAvJKK1uLMMAv1uzAv  1c{1a& KLLq A!%AqQQ1 5HA"Ava 1q5\ S1R4Z1  6 Q]  rc4\^R^4FpV^,'dV)p\W4pVR8Xd\V^,4p\V^^V, ^,V^,V, 4wrEpV^8XgV^8XdR#\V^, 4F8pWU,^V,, V,pV^8XdR#\ V^V4pK: R#V^8XdW,'dR#V^ 8XgK\ V4'gKR# \ R4h)rJrwTFzDappropriate value for D cannot be found in is_strong_selfridge_prp()r)r`rMrrorBr@r)r rkrLr0rlrmrnras& r_is_strong_selfridge_prpr7s 1i # q55A 1L 7!a% A&q!acaZ!a%AFHA"Ava1q5\S1R4Z1$6Q] "  6aee 7y||'$( [ \\rctV^8d \R4hV^8XdR#V^,^8XdV^8H#\V4#)rz8is_strong_selfridge_prp() requires 'n' be greater than 0F)rrr4s&ris_strong_selfridge_prprOs>1uSTTAv1uzAv #A &&rcV^8d \R4hV^8XdR#V^,^8XdV^8H#\V^4;'d \V4#)rz,is_bpsw_prp() requires 'n' be greater than 0F)rrbrxr4s&r is_bpsw_prprYsN1uGHHAv1uzAv !Q  8 8$5a$88rcV^8d \R4hV^8XdR#V^,^8XdV^8H#\V^4;'d \V4#)rz3is_strong_bpsw_prp() requires 'n' be greater than 0F)rrbrr4s&ris_strong_bpsw_prprcsN1uNOOAv1uzAv !Q  ? ?$rs  #) q!A/0cQ1q5\.BOAF*aAEl*+ )@&8" #   hh hh '*!(H >   0 +\ ! 1   |~ 3 >%VP 2]0'9@r