from math import gcd from sympy.ntheory.generate import Sieve, sieve from sympy.ntheory.primetest import (mr, _lucas_extrastrong_params, is_lucas_prp, is_square, is_strong_lucas_prp, is_extra_strong_lucas_prp, proth_test, isprime, is_euler_pseudoprime, is_gaussian_prime, is_fermat_pseudoprime, is_euler_jacobi_pseudoprime, MERSENNE_PRIME_EXPONENTS, _lucas_lehmer_primality_test, is_mersenne_prime) from sympy.testing.pytest import slow, raises from sympy.core.numbers import I, Float def test_is_fermat_pseudoprime(): assert is_fermat_pseudoprime(5, 1) assert is_fermat_pseudoprime(9, 1) def test_euler_pseudoprimes(): assert is_euler_pseudoprime(13, 1) assert is_euler_pseudoprime(15, 1) assert is_euler_pseudoprime(17, 6) assert is_euler_pseudoprime(101, 7) assert is_euler_pseudoprime(1009, 10) assert is_euler_pseudoprime(11287, 41) raises(ValueError, lambda: is_euler_pseudoprime(0, 4)) raises(ValueError, lambda: is_euler_pseudoprime(3, 0)) raises(ValueError, lambda: is_euler_pseudoprime(15, 6)) # A006970 euler_prp = [341, 561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 5461, 6601, 8321, 8481, 10261, 10585] for p in euler_prp: assert is_euler_pseudoprime(p, 2) # A048950 euler_prp = [121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009] for p in euler_prp: assert is_euler_pseudoprime(p, 3) # A033181 absolute_euler_prp = [1729, 2465, 15841, 41041, 46657, 75361, 162401, 172081, 399001, 449065, 488881] for p in absolute_euler_prp: for a in range(2, p): if gcd(a, p) != 1: continue assert is_euler_pseudoprime(p, a) def test_is_euler_jacobi_pseudoprime(): assert is_euler_jacobi_pseudoprime(11, 1) assert is_euler_jacobi_pseudoprime(15, 1) def test_lucas_extrastrong_params(): assert _lucas_extrastrong_params(3) == (5, 3, 1) assert _lucas_extrastrong_params(5) == (12, 4, 1) assert _lucas_extrastrong_params(7) == (5, 3, 1) assert _lucas_extrastrong_params(9) == (0, 0, 0) assert _lucas_extrastrong_params(11) == (21, 5, 1) assert _lucas_extrastrong_params(59) == (32, 6, 1) assert _lucas_extrastrong_params(479) == (117, 11, 1) def test_is_extra_strong_lucas_prp(): assert is_extra_strong_lucas_prp(4) == False assert is_extra_strong_lucas_prp(989) == True assert is_extra_strong_lucas_prp(10877) == True assert is_extra_strong_lucas_prp(9) == False assert is_extra_strong_lucas_prp(16) == False assert is_extra_strong_lucas_prp(169) == False @slow def test_prps(): oddcomposites = [n for n in range(1, 10**5) if n % 2 and not isprime(n)] # A checksum would be better. assert sum(oddcomposites) == 2045603465 assert [n for n in oddcomposites if mr(n, [2])] == [ 2047, 3277, 4033, 4681, 8321, 15841, 29341, 42799, 49141, 52633, 65281, 74665, 80581, 85489, 88357, 90751] assert [n for n in oddcomposites if mr(n, [3])] == [ 121, 703, 1891, 3281, 8401, 8911, 10585, 12403, 16531, 18721, 19345, 23521, 31621, 44287, 47197, 55969, 63139, 74593, 79003, 82513, 87913, 88573, 97567] assert [n for n in oddcomposites if mr(n, [325])] == [ 9, 25, 27, 49, 65, 81, 325, 341, 343, 697, 1141, 2059, 2149, 3097, 3537, 4033, 4681, 4941, 5833, 6517, 7987, 8911, 12403, 12913, 15043, 16021, 20017, 22261, 23221, 24649, 24929, 31841, 35371, 38503, 43213, 44173, 47197, 50041, 55909, 56033, 58969, 59089, 61337, 65441, 68823, 72641, 76793, 78409, 85879] assert not any(mr(n, [9345883071009581737]) for n in oddcomposites) assert [n for n in oddcomposites if is_lucas_prp(n)] == [ 323, 377, 1159, 1829, 3827, 5459, 5777, 9071, 9179, 10877, 11419, 11663, 13919, 14839, 16109, 16211, 18407, 18971, 19043, 22499, 23407, 24569, 25199, 25877, 26069, 27323, 32759, 34943, 35207, 39059, 39203, 39689, 40309, 44099, 46979, 47879, 50183, 51983, 53663, 56279, 58519, 60377, 63881, 69509, 72389, 73919, 75077, 77219, 79547, 79799, 82983, 84419, 86063, 90287, 94667, 97019, 97439] assert [n for n in oddcomposites if is_strong_lucas_prp(n)] == [ 5459, 5777, 10877, 16109, 18971, 22499, 24569, 25199, 40309, 58519, 75077, 97439] assert [n for n in oddcomposites if is_extra_strong_lucas_prp(n) ] == [ 989, 3239, 5777, 10877, 27971, 29681, 30739, 31631, 39059, 72389, 73919, 75077] def test_proth_test(): # Proth number A080075 = [3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129, 145, 161, 177, 193] # Proth prime A080076 = [3, 5, 13, 17, 41, 97, 113, 193] for n in range(200): if n in A080075: assert proth_test(n) == (n in A080076) else: raises(ValueError, lambda: proth_test(n)) def test_lucas_lehmer_primality_test(): for p in sieve.primerange(3, 100): assert _lucas_lehmer_primality_test(p) == (p in MERSENNE_PRIME_EXPONENTS) def test_is_mersenne_prime(): assert is_mersenne_prime(-3) is False assert is_mersenne_prime(3) is True assert is_mersenne_prime(10) is False assert is_mersenne_prime(127) is True assert is_mersenne_prime(511) is False assert is_mersenne_prime(131071) is True assert is_mersenne_prime(2147483647) is True def test_isprime(): s = Sieve() s.extend(100000) ps = set(s.primerange(2, 100001)) for n in range(100001): # if (n in ps) != isprime(n): print n assert (n in ps) == isprime(n) assert isprime(179424673) assert isprime(20678048681) assert isprime(1968188556461) assert isprime(2614941710599) assert isprime(65635624165761929287) assert isprime(1162566711635022452267983) assert isprime(77123077103005189615466924501) assert isprime(3991617775553178702574451996736229) assert isprime(273952953553395851092382714516720001799) assert isprime(int(''' 531137992816767098689588206552468627329593117727031923199444138200403\ 559860852242739162502265229285668889329486246501015346579337652707239\ 409519978766587351943831270835393219031728127''')) # Some Mersenne primes assert isprime(2**61 - 1) assert isprime(2**89 - 1) assert isprime(2**607 - 1) # (but not all Mersenne's are primes assert not isprime(2**601 - 1) # pseudoprimes #------------- # to some small bases assert not isprime(2152302898747) assert not isprime(3474749660383) assert not isprime(341550071728321) assert not isprime(3825123056546413051) # passes the base set [2, 3, 7, 61, 24251] assert not isprime(9188353522314541) # large examples assert not isprime(877777777777777777777777) # conjectured psi_12 given at http://mathworld.wolfram.com/StrongPseudoprime.html assert not isprime(318665857834031151167461) # conjectured psi_17 given at http://mathworld.wolfram.com/StrongPseudoprime.html assert not isprime(564132928021909221014087501701) # Arnault's 1993 number; a factor of it is # 400958216639499605418306452084546853005188166041132508774506\ # 204738003217070119624271622319159721973358216316508535816696\ # 9145233813917169287527980445796800452592031836601 assert not isprime(int(''' 803837457453639491257079614341942108138837688287558145837488917522297\ 427376533365218650233616396004545791504202360320876656996676098728404\ 396540823292873879185086916685732826776177102938969773947016708230428\ 687109997439976544144845341155872450633409279022275296229414984230688\ 1685404326457534018329786111298960644845216191652872597534901''')) # Arnault's 1995 number; can be factored as # p1*(313*(p1 - 1) + 1)*(353*(p1 - 1) + 1) where p1 is # 296744956686855105501541746429053327307719917998530433509950\ # 755312768387531717701995942385964281211880336647542183455624\ # 93168782883 assert not isprime(int(''' 288714823805077121267142959713039399197760945927972270092651602419743\ 230379915273311632898314463922594197780311092934965557841894944174093\ 380561511397999942154241693397290542371100275104208013496673175515285\ 922696291677532547504444585610194940420003990443211677661994962953925\ 045269871932907037356403227370127845389912612030924484149472897688540\ 6024976768122077071687938121709811322297802059565867''')) sieve.extend(3000) assert isprime(2819) assert not isprime(2931) raises(ValueError, lambda: isprime(2.0)) raises(ValueError, lambda: isprime(Float(2))) def test_is_square(): assert [i for i in range(25) if is_square(i)] == [0, 1, 4, 9, 16] # issue #17044 assert not is_square(60 ** 3) assert not is_square(60 ** 5) assert not is_square(84 ** 7) assert not is_square(105 ** 9) assert not is_square(120 ** 3) def test_is_gaussianprime(): assert is_gaussian_prime(7*I) assert is_gaussian_prime(7) assert is_gaussian_prime(2 + 3*I) assert not is_gaussian_prime(2 + 2*I)