U 9%eQ@sdZddlmZddlmZddlmZddlmZddl m Z ddl m Z dd Zd d Zd$d dZddZddZddZddZddZddZddZddZdd Zd!d"Zd#S)%z Primality testing )igcd)integer_nthroot)sympify)HAS_GMPY)as_int)bitcountcGstdd|DS)Ncss|]}t|VqdS)N)int).0_r V/var/www/html/Darija-Ai-API/env/lib/python3.8/site-packages/sympy/ntheory/primetest.py sz_int_tuple..)tuple)ir r r _int_tuplesrcCszddlm}t||gsdSt|}|d}t||||?|}|dkrLdS||dkr\dSt|d|}|dkrLdSqLdS)afReturns True if n is prime or an Euler pseudoprime to base b, else False. Euler Pseudoprime : In arithmetic, an odd composite integer n is called an euler pseudoprime to base a, if a and n are coprime and satisfy the modular arithmetic congruence relation : a ^ (n-1)/2 = + 1(mod n) or a ^ (n-1)/2 = - 1(mod n) (where mod refers to the modulo operation). Examples ======== >>> from sympy.ntheory.primetest import is_euler_pseudoprime >>> is_euler_pseudoprime(2, 5) True References ========== .. [1] https://en.wikipedia.org/wiki/Euler_pseudoprime rtrailingFTN)sympy.ntheory.factor_rmrrpow)nbrrcr r r is_euler_pseudoprimes    rTcCs|r$t|}|dkrdS|dkr$dSdd|d@>@s8dS|d}d d|d >@sTdSd d|d >@shdSd d|d>@s|dSt|ddS)aReturn True if n == a * a for some integer a, else False. If n is suspected of *not* being a square then this is a quick method of confirming that it is not. Examples ======== >>> from sympy.ntheory.primetest import is_square >>> is_square(25) True >>> is_square(2) False References ========== .. [1] https://mersenneforum.org/showpost.php?p=110896 See Also ======== sympy.core.power.integer_nthroot rF)rrTl H0!@@riE l$ clBEa2P 0@\ [lS qpDQBUr)rr)rprepmr r r is_square?s   r#cCsdt|||}|dks ||dkr$dStd|D]0}t|d|}||dkrPdS|dkr.dSq.dS)zMiller-Rabin strong pseudoprime test for one base. Return False if n is definitely composite, True if n is probably prime, with a probability greater than 3/4. rTrF)rrange)rbasestrjr r r _test~s   r)cCsddlm}ddlm}t|}|dkr,dS||d}||?}|D]8}||krX||;}|dkrD||}t||||sDdSqDdS)aGPerform a Miller-Rabin strong pseudoprime test on n using a given list of bases/witnesses. References ========== .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers: A Computational Perspective", Springer, 2nd edition, 135-138 A list of thresholds and the bases they require are here: https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test#Deterministic_variants Examples ======== >>> from sympy.ntheory.primetest import mr >>> mr(1373651, [2, 3]) False >>> mr(479001599, [31, 73]) True rr)ZZrFrT)rrZsympy.polys.domainsr*rr))rbasesrr*r&r'r%r r r rs   rc Cs||d|}|dkr td|dkr0td|dkr@td|dkrTtdd|Sd}|}|}t|}|dkr|dkr|||}||d|}|d8}||d?d@rr|||||||}}|d@r||7}|d@r||7}|d?|d?}}qrn~|dkr|dkr|dkr|||}|dkrL||d|}n||d|}d}|d8}||d?d@r|||||}}|d@r||7}|d@r||7}|d?|d?}}d}qn|dkr|||}||d||}||9}|d8}||d?d@rx|||||||}}|d@rL||7}|d@r^||7}|d?|d?}}||9}||;}qt|||||S) aReturn the modular Lucas sequence (U_k, V_k, Q_k). 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. The modular Lucas sequences are used in numerous places in number theory, especially in the Lucas compositeness tests and the various n + 1 proofs. Examples ======== >>> from sympy.ntheory.primetest import _lucas_sequence >>> N = 10**2000 + 4561 >>> sol = U, V, Qk = _lucas_sequence(N, 3, 1, N//2); sol (0, 2, 1) rzn must be >= 2rzk must be >= 0zD must not be zeror) ValueErrorr _bitlength) rPQkDUVQkrr r r _lucas_sequencesp               r7cCsvddlm}d}tt||}|dkr2||kr2dS|||dkrBqb|dkrV| d}q| d}qt|dd|dS) a Calculates the Selfridge parameters (D, P, Q) for n. This is method A from page 1401 of Baillie and Wagstaff. References ========== .. [1] "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf r jacobi_symbolrrrrr-rr,)sympy.ntheory.residue_ntheoryr9rabsr)rr9r3gr r r _lucas_selfridge_paramss   r?cCsfddlm}d\}}}t||}|dkr4||kr4dS|||dkrDqZ|d7}||d}qt|||S)zCalculates the "extra strong" parameters (D, P, Q) for n. References ========== .. [1] OEIS A217719: Extra Strong Lucas Pseudoprimes https://oeis.org/A217719 .. [1] https://en.wikipedia.org/wiki/Lucas_pseudoprime rr8)rr:rr;r-r,)r<r9rr)rr9r0r1r3r>r r r _lucas_extrastrong_params*s   rAcCstt|}|dkrdS|dks(|ddkr,dSt|dr:dSt|\}}}|dkrTdSt||||d\}}}|dkS)aStandard Lucas compositeness test with Selfridge parameters. Returns False if n is definitely composite, and True if n is a Lucas probable prime. This is typically used in combination with the Miller-Rabin test. References ========== - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - OEIS A217120: Lucas Pseudoprimes https://oeis.org/A217120 - https://en.wikipedia.org/wiki/Lucas_pseudoprime Examples ======== >>> from sympy.ntheory.primetest import isprime, is_lucas_prp >>> for i in range(10000): ... if is_lucas_prp(i) and not isprime(i): ... print(i) 323 377 1159 1829 3827 5459 5777 9071 9179 rTrFr)rr#r?r7)rr3r0r1r4r5r6r r r is_lucas_prp@s  rBc Csddlm}t|}|dkr dS|dks4|ddkr8dSt|drFdSt|\}}}|dkr`dS||d}|d|?}t||||\}}} |dks|dkrdStd|D]2} ||d| |}|dkrdSt| d|} qdS)aStrong Lucas compositeness test with Selfridge parameters. Returns False if n is definitely composite, and True if n is a strong Lucas probable prime. This is often used in combination with the Miller-Rabin test, and in particular, when combined with M-R base 2 creates the strong BPSW test. References ========== - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - OEIS A217255: Strong Lucas Pseudoprimes https://oeis.org/A217255 - https://en.wikipedia.org/wiki/Lucas_pseudoprime - https://en.wikipedia.org/wiki/Baillie-PSW_primality_test Examples ======== >>> from sympy.ntheory.primetest import isprime, is_strong_lucas_prp >>> for i in range(20000): ... if is_strong_lucas_prp(i) and not isprime(i): ... print(i) 5459 5777 10877 16109 18971 rrrTFr)rrrr#r?r7r$r rrr3r0r1r&r2r4r5r6rr r r is_strong_lucas_prpos,    rDc Csddlm}t|}|dkr dS|dks4|ddkr8dSt|drFdSt|\}}}|dkr`dS||d}|d|?}t||||\}}} |dkr|dks||dkrdStd|D]"} |dkrdS||d|}qdS)atExtra Strong Lucas compositeness test. Returns False if n is definitely composite, and True if n is a "extra strong" Lucas probable prime. The parameters are selected using P = 3, Q = 1, then incrementing P until (D|n) == -1. The test itself is as defined in Grantham 2000, from the Mo and Jones preprint. The parameter selection and test are the same as used in OEIS A217719, Perl's Math::Prime::Util, and the Lucas pseudoprime page on Wikipedia. With these parameters, there are no counterexamples below 2^64 nor any known above that range. It is 20-50% faster than the strong test. Because of the different parameters selected, there is no relationship between the strong Lucas pseudoprimes and extra strong Lucas pseudoprimes. In particular, one is not a subset of the other. References ========== - "Frobenius Pseudoprimes", Jon Grantham, 2000. https://www.ams.org/journals/mcom/2001-70-234/S0025-5718-00-01197-2/ - OEIS A217719: Extra Strong Lucas Pseudoprimes https://oeis.org/A217719 - https://en.wikipedia.org/wiki/Lucas_pseudoprime Examples ======== >>> from sympy.ntheory.primetest import isprime, is_extra_strong_lucas_prp >>> for i in range(20000): ... if is_extra_strong_lucas_prp(i) and not isprime(i): ... print(i) 989 3239 5777 10877 rrrTFr)rrrr#rAr7r$rCr r r is_extra_strong_lucas_prps*-    rEcCsz t|}Wntk r"YdSX|dkr0dS|dks\|ddks\|ddks\|ddkr`dS|dkrldS|d dks|d dks|d dks|d dks|d dks|ddks|ddks|ddks|ddks|ddks|ddks|ddkrdS|dkr&dS|dkrJtd||dkoH|dkSddlm}||jdkr|||\}}||kStdkrddlm }m }||do||S|dkrt |dgS|dkrt |ddgS|d krt |d!d"d#gS|d$krt |dd%d&d'gS|d(kr,t |dd)d*d+d,gS|d-krLt |dd.d/d0d1d2gS|d3krnt |dd4d5d6d7d8d9gS|d:krt |dddd d d d d ddddg S|d;krt |dddd d d d d dddddg St |dgot |S)>> from sympy.ntheory import isprime >>> isprime(13) True >>> isprime(13.0) # limited precision False >>> isprime(15) False Notes ===== This routine is intended only for integer input, not numerical expressions which may represent numbers. Floats are also rejected as input because they represent numbers of limited precision. While it is tempting to permit 7.0 to represent an integer there are errors that may "pass silently" if this is allowed: >>> from sympy import Float, S >>> int(1e3) == 1e3 == 10**3 True >>> int(1e23) == 1e23 True >>> int(1e23) == 10**23 False >>> near_int = 1 + S(1)/10**19 >>> near_int == int(near_int) False >>> n = Float(near_int, 10) # truncated by precision >>> n == int(n) True >>> n = Float(near_int, 20) >>> n == int(n) False See Also ======== sympy.ntheory.generate.primerange : Generates all primes in a given range sympy.ntheory.generate.primepi : Return the number of primes less than or equal to n sympy.ntheory.generate.prime : Return the nth prime References ========== - https://en.wikipedia.org/wiki/Strong_pseudoprime - "Lucas Pseudoprimes", Baillie and Wagstaff, 1980. http://mpqs.free.fr/LucasPseudoprimes.pdf - https://en.wikipedia.org/wiki/Baillie-PSW_primality_test F)rr@r:Trrr@r:1 %)+/i iz)ii i5i!IiMiQ[)siever-) is_strong_prpis_strong_selfridge_prpi6lj_| i94l85Plp8K0T l?_Fl|a?bUl$|3r_ l(P7lk^AHl@0le7:`V!l"SV=Q-O lv-lmorl\^E~8l)VmAlvKHu8l~_!@l\]AlbY+5l0PrlMK Tl0KOM liEi$inii=ikl7 y_@I7l%!Hn fW) rr.rZsympy.ntheory.generaterSZ_listsearchrZgmpy2rTrUrrD)rr&lurTrUr r r isprimestE ,8                     " $%rYcCst|}|\}}t|dd}t|dd}|dkrPt|}t|oN|ddkS|dkrtt|}t|or|ddkSt|d|dS)z}Test if num is a Gaussian prime number. References ========== .. [1] https://oeis.org/wiki/Gaussian_primes F)strictrr,r@r)rZ as_real_imagrr=rY)numarr r r is_gaussian_primes    r]N)T)__doc__Zsympy.core.numbersrZsympy.core.powerrZsympy.core.sympifyrZsympy.external.gmpyrZsympy.utilities.miscrZ mpmath.libmprr/rrr#r)rr7r?rArBrDrErYr]r r r r s(      , ?+T/;I3