def test_totient():
assert [totient(k) for k in range(1, 12)] == \
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10]
assert totient(5005) == 2880
assert totient(5006) == 2502
assert totient(5009) == 5008
assert totient(2**100) == 2**99
m = Symbol("m", integer=True)
assert totient(m)
assert totient(m).subs({m: 3**10}) == 3**10 - 3**9
assert summation(totient(m), (m, 1, 11)) == 42
n = Symbol("n", integer=True, positive=True)
assert totient(n).is_integer
assert totient(m).is_integer is None
def test_totient():
assert [totient(k) for k in range(1, 12)] == \
[1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10]
assert totient(5005) == 2880
assert totient(5006) == 2502
assert totient(5009) == 5008
assert totient(2**100) == 2**99
m = Symbol("m", integer=True)
assert totient(m)
assert totient(m).subs({m: 3**10}) == 3**10 - 3**9
assert summation(totient(m), (m, 1, 11)) == 42
n = Symbol("n", integer=True, positive=True)
assert totient(n).is_integer
assert totient(m).is_integer is None
def test_generate():
assert nextprime(-4) == 2
assert nextprime(2) == 3
assert nextprime(5) == 7
assert nextprime(12) == 13
assert nextprime(90) == 97
assert nextprime(10**40) == (10**40 + 121)
pytest.raises(ValueError, lambda: prevprime(2))
assert prevprime(3) == 2
assert prevprime(7) == 5
assert prevprime(13) == 11
assert prevprime(97) == 89
assert prevprime(10**40) == (10**40 - 17)
assert list(primerange(3001110000000, 799999999)) == []
assert list(primerange(2, 7)) == [2, 3, 5]
assert list(primerange(2, 10)) == [2, 3, 5, 7]
assert list(primerange(
1050, 1100)) == [1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097]
s = Sieve()
for i in range(30, 2350, 376):
for j in range(2, 5096, 1139):
A = list(s.primerange(i, i + j))
B = list(primerange(i, i + j))
assert A == B
s = Sieve()
assert s[10] == 29
assert nextprime(2, 2) == 5
pytest.raises(ValueError, lambda: totient(0))
pytest.raises(ValueError, lambda: primorial(0))
assert mr(1, [2]) is False
def func(i):
return (i**2 + 1) % 51
assert next(cycle_length(func, 4)) == (6, 2)
assert list(cycle_length(func, 4, values=True)) == \
[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
assert next(cycle_length(func, 4, nmax=5)) == (5, None)
assert list(cycle_length(func, 4, nmax=5, values=True)) == \
[17, 35, 2, 5, 26]
assert list(cycle_length(lambda i: 0, 0, nmax=0)) == [(1, 0)]
def test_generate():
assert nextprime(-4) == 2
assert nextprime(2) == 3
assert nextprime(5) == 7
assert nextprime(12) == 13
assert nextprime(90) == 97
assert nextprime(10**40) == (10**40 + 121)
pytest.raises(ValueError, lambda: prevprime(2))
assert prevprime(3) == 2
assert prevprime(7) == 5
assert prevprime(13) == 11
assert prevprime(97) == 89
assert prevprime(10**40) == (10**40 - 17)
assert list(primerange(3001110000000, 799999999)) == []
assert list(primerange(2, 7)) == [2, 3, 5]
assert list(primerange(2, 10)) == [2, 3, 5, 7]
assert list(primerange(1050, 1100)) == [1051, 1061,
1063, 1069, 1087, 1091, 1093, 1097]
s = Sieve()
for i in range(30, 2350, 376):
for j in range(2, 5096, 1139):
A = list(s.primerange(i, i + j))
B = list(primerange(i, i + j))
assert A == B
s = Sieve()
assert s[10] == 29
assert nextprime(2, 2) == 5
pytest.raises(ValueError, lambda: totient(0))
pytest.raises(ValueError, lambda: primorial(0))
assert mr(1, [2]) is False
def func(i):
return (i**2 + 1) % 51
assert next(cycle_length(func, 4)) == (6, 2)
assert list(cycle_length(func, 4, values=True)) == \
[17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14]
assert next(cycle_length(func, 4, nmax=5)) == (5, None)
assert list(cycle_length(func, 4, nmax=5, values=True)) == \
[17, 35, 2, 5, 26]
assert list(cycle_length(lambda i: 0, 0, nmax=0)) == [(1, 0)]
def test_residue():
assert n_order(2, 13) == 12
assert [n_order(a, 7) for a in range(1, 7)] == \
[1, 3, 6, 3, 6, 2]
assert n_order(5, 17) == 16
assert n_order(17, 11) == n_order(6, 11)
assert n_order(101, 119) == 6
assert n_order(11, (10**50 + 151)**2) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650
pytest.raises(ValueError, lambda: n_order(6, 9))
assert is_primitive_root(2, 7) is False
assert is_primitive_root(3, 8) is False
assert is_primitive_root(11, 14) is False
assert is_primitive_root(12, 17) == is_primitive_root(29, 17)
pytest.raises(ValueError, lambda: is_primitive_root(3, 6))
assert [primitive_root(i) for i in range(2, 31)] == [1, 2, 3, 2, 5, 3,
None, 2, 3, 2, None, 2, 3, None, None, 3, 5, 2, None, None, 7, 5,
None, 2, 7, 2, None, 2, None]
for p in primerange(3, 100):
it = _primitive_root_prime_iter(p)
assert len(list(it)) == totient(totient(p))
assert primitive_root(97) == 5
assert primitive_root(97**2) == 5
assert primitive_root(40487) == 5
# note that primitive_root(40487) + 40487 = 40492 is a primitive root
# of 40487**2, but it is not the smallest
assert primitive_root(40487**2) == 10
assert primitive_root(82) == 7
p = 10**50 + 151
assert primitive_root(p) == 11
assert primitive_root(2*p) == 11
assert primitive_root(p**2) == 11
pytest.raises(ValueError, lambda: primitive_root(-3))
assert is_quad_residue(3, 7) is False
assert is_quad_residue(10, 13) is True
assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139)
assert is_quad_residue(207, 251) is True
assert is_quad_residue(0, 1) is True
assert is_quad_residue(1, 1) is True
assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True
assert is_quad_residue(1, 4) is True
assert is_quad_residue(2, 27) is False
assert is_quad_residue(13122380800, 13604889600) is True
assert [j for j in range(14) if is_quad_residue(j, 14)] == \
[0, 1, 2, 4, 7, 8, 9, 11]
pytest.raises(ValueError, lambda: is_quad_residue(1.1, 2))
pytest.raises(ValueError, lambda: is_quad_residue(2, 0))
assert quadratic_residues(12) == [0, 1, 4, 9]
assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12]
assert [len(quadratic_residues(i)) for i in range(1, 20)] == \
[1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10]
assert list(sqrt_mod_iter(6, 2)) == [0]
assert sqrt_mod(3, 13) == 4
assert sqrt_mod(3, -13) == 4
assert sqrt_mod(6, 23) == 11
assert sqrt_mod(345, 690) == 345
assert sqrt_mod(9, 18) == 3
for p in range(3, 100):
d = collections.defaultdict(list)
for i in range(p):
d[pow(i, 2, p)].append(i)
for i in range(1, p):
it = sqrt_mod_iter(i, p)
v = sqrt_mod(i, p, True)
if v:
v = sorted(v)
assert d[i] == v
else:
assert not d[i]
assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24]
assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78]
assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240]
assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72]
assert sqrt_mod(81, 3**5, True) == [9, 18, 36, 45, 63, 72, 90, 99, 117,
126, 144, 153, 171, 180, 198, 207, 225, 234]
assert sqrt_mod(81, 3**6, True) == [9, 72, 90, 153, 171, 234, 252, 315,
333, 396, 414, 477, 495, 558, 576, 639, 657, 720]
assert sqrt_mod(81, 3**7, True) == [9, 234, 252, 477, 495, 720, 738, 963,
981, 1206, 1224, 1449, 1467, 1692, 1710, 1935, 1953, 2178]
for a, p in [(26214400, 32768000000), (26214400, 16384000000),
(262144, 1048576), (87169610025, 163443018796875),
(22315420166400, 167365651248000000)]:
assert pow(sqrt_mod(a, p), 2, p) == a
n = 70
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+2)
it = sqrt_mod_iter(a, p)
for i in range(10):
assert pow(next(it), 2, p) == a
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+3)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
n = 100
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+1)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
assert type(next(sqrt_mod_iter(9, 27))) is int
assert isinstance(next(sqrt_mod_iter(9, 27, ZZ)), type(ZZ(1)))
assert isinstance(next(sqrt_mod_iter(1, 7, ZZ)), type(ZZ(1)))
assert list(sqrt_mod_iter(4, 919, ZZ)) == [2, 917]
assert list(sqrt_mod_iter(6, 146, ZZ)) == [88, 58]
pytest.raises(ValueError, lambda: is_nthpow_residue(+2, +1, 0))
pytest.raises(ValueError, lambda: is_nthpow_residue(+2, -1, 5))
pytest.raises(ValueError, lambda: is_nthpow_residue(-2, +1, 5))
assert is_nthpow_residue(2, 1, 5)
assert is_nthpow_residue(1, 0, 1) is False
assert is_nthpow_residue(1, 0, 2) is True
assert is_nthpow_residue(3, 0, 2) is False
assert is_nthpow_residue(0, 1, 8) is True
assert is_nthpow_residue(2, 3, 2) is False
assert is_nthpow_residue(2, 3, 9) is False
assert is_nthpow_residue(3, 5, 30) is True
assert is_nthpow_residue(21, 11, 20) is True
assert is_nthpow_residue(7, 10, 20) is False
assert is_nthpow_residue(5, 10, 20) is True
assert is_nthpow_residue(3, 10, 48) is False
assert is_nthpow_residue(1, 10, 40) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(1, 10, 24) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(2, 10, 48) is False
assert is_nthpow_residue(81, 3, 972) is False
assert is_nthpow_residue(243, 5, 5103) is True
assert is_nthpow_residue(243, 3, 1240029) is False
assert {a for a in range(1024)
if is_nthpow_residue(a, 56, 1024)} == {pow(i, 56, 1024)
for i in range(1024)}
assert {a for a in range(2048)
if is_nthpow_residue(a, 256, 2048)} == {pow(i, 256, 2048)
for i in range(2048)}
x = {pow(i, 11, 324000) for i in range(1000)}
assert [is_nthpow_residue(a, 11, 324000) for a in x]
x = {pow(i, 17, 22217575536) for i in range(1000)}
assert [is_nthpow_residue(a, 17, 22217575536) for a in x]
assert is_nthpow_residue(676, 3, 5364)
assert is_nthpow_residue(9, 12, 36)
assert is_nthpow_residue(32, 10, 41)
assert is_nthpow_residue(4, 2, 64)
assert is_nthpow_residue(31, 4, 41)
assert not is_nthpow_residue(2, 2, 5)
assert is_nthpow_residue(8547, 12, 10007)
assert nthroot_mod(1801, 11, 2663) == 44
for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663),
(26118163, 1303, 33333347), (1499, 7, 2663), (595, 6, 2663),
(1714, 12, 2663), (28477, 9, 33343)]:
r = nthroot_mod(a, q, p)
assert pow(r, q, p) == a
assert nthroot_mod(11, 3, 109) is None
assert nthroot_mod(6, 12, 5) == 1
pytest.raises(NotImplementedError, lambda: nthroot_mod(16, 5, 36))
pytest.raises(NotImplementedError, lambda: nthroot_mod(9, 16, 36))
for p in primerange(5, 100):
qv = range(3, p, 4)
for q in qv:
d = collections.defaultdict(list)
for i in range(p):
d[pow(i, q, p)].append(i)
for a in range(1, p - 1):
res = nthroot_mod(a, q, p, True)
if d[a]:
assert d[a] == res
else:
assert res is None
assert legendre_symbol(5, 11) == 1
assert legendre_symbol(25, 41) == 1
assert legendre_symbol(67, 101) == -1
assert legendre_symbol(0, 13) == 0
assert legendre_symbol(9, 3) == 0
pytest.raises(ValueError, lambda: legendre_symbol(2, 4))
assert jacobi_symbol(25, 41) == 1
assert jacobi_symbol(-23, 83) == -1
assert jacobi_symbol(3, 9) == 0
assert jacobi_symbol(42, 97) == -1
assert jacobi_symbol(3, 5) == -1
assert jacobi_symbol(7, 9) == 1
assert jacobi_symbol(0, 3) == 0
assert jacobi_symbol(0, 1) == 1
assert jacobi_symbol(2, 1) == 1
assert jacobi_symbol(1, 3) == 1
pytest.raises(ValueError, lambda: jacobi_symbol(3, 8))
assert mobius(13*7) == 1
assert mobius(1) == 1
assert mobius(13*7*5) == -1
assert mobius(13**2) == 0
pytest.raises(ValueError, lambda: mobius(-3))
p = Symbol('p', integer=True, positive=True, prime=True)
x = Symbol('x', positive=True)
i = Symbol('i', integer=True)
assert mobius(p) == -1
pytest.raises(TypeError, lambda: mobius(x))
pytest.raises(ValueError, lambda: mobius(i))
mobius(Symbol('p', positive=True, integer=True))
assert _discrete_log_trial_mul(41, 15, 7) == 3
assert _discrete_log_trial_mul(587, 2**7, 2) == 7
assert _discrete_log_trial_mul(941, 7**18, 7) == 18
assert _discrete_log_trial_mul(389, 3**81, 3) == 81
assert _discrete_log_trial_mul(191, 19**123, 19) == 123
pytest.raises(ValueError, lambda: _discrete_log_trial_mul(11, 7, 31))
assert _discrete_log_shanks_steps(41, 15, 7) == 3
assert _discrete_log_shanks_steps(442879, 7**2, 7) == 2
assert _discrete_log_shanks_steps(874323, 5**19, 5) == 19
assert _discrete_log_shanks_steps(6876342, 7**71, 7) == 71
assert _discrete_log_shanks_steps(2456747, 3**321, 3) == 321
pytest.raises(ValueError, lambda: _discrete_log_shanks_steps(11, 7, 31))
assert _discrete_log_pohlig_hellman(251, 210, 71) == 197
assert _discrete_log_pohlig_hellman(98376431, 11**9, 11) == 9
assert _discrete_log_pohlig_hellman(78723213, 11**31, 11) == 31
assert _discrete_log_pohlig_hellman(32942478, 11**98, 11) == 98
assert _discrete_log_pohlig_hellman(14789363, 11**444, 11) == 444
pytest.raises(ValueError, lambda: _discrete_log_pohlig_hellman(11, 7, 31))
assert discrete_log(587, 2**9, 2) == 9
assert discrete_log(2456747, 3**51, 3) == 51
assert discrete_log(32942478, 11**127, 11) == 127
assert discrete_log(432751500361, 7**324, 7) == 324
pytest.raises(ValueError, lambda: discrete_log(11, 7, 31))
def test_residue():
assert n_order(2, 13) == 12
assert [n_order(a, 7) for a in range(1, 7)] == \
[1, 3, 6, 3, 6, 2]
assert n_order(5, 17) == 16
assert n_order(17, 11) == n_order(6, 11)
assert n_order(101, 119) == 6
assert n_order(11, (10**50 + 151)**2) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650
pytest.raises(ValueError, lambda: n_order(6, 9))
assert is_primitive_root(2, 7) is False
assert is_primitive_root(3, 8) is False
assert is_primitive_root(11, 14) is False
assert is_primitive_root(12, 17) == is_primitive_root(29, 17)
pytest.raises(ValueError, lambda: is_primitive_root(3, 6))
assert [primitive_root(i) for i in range(2, 31)] == [1, 2, 3, 2, 5, 3,
None, 2, 3, 2, None, 2, 3, None, None, 3, 5, 2, None, None, 7, 5,
None, 2, 7, 2, None, 2, None]
for p in primerange(3, 100):
it = _primitive_root_prime_iter(p)
assert len(list(it)) == totient(totient(p))
assert primitive_root(97) == 5
assert primitive_root(97**2) == 5
assert primitive_root(40487) == 5
# note that primitive_root(40487) + 40487 = 40492 is a primitive root
# of 40487**2, but it is not the smallest
assert primitive_root(40487**2) == 10
assert primitive_root(82) == 7
p = 10**50 + 151
assert primitive_root(p) == 11
assert primitive_root(2*p) == 11
assert primitive_root(p**2) == 11
pytest.raises(ValueError, lambda: primitive_root(-3))
assert is_quad_residue(3, 7) is False
assert is_quad_residue(10, 13) is True
assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139)
assert is_quad_residue(207, 251) is True
assert is_quad_residue(0, 1) is True
assert is_quad_residue(1, 1) is True
assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True
assert is_quad_residue(1, 4) is True
assert is_quad_residue(2, 27) is False
assert is_quad_residue(13122380800, 13604889600) is True
assert [j for j in range(14) if is_quad_residue(j, 14)] == \
[0, 1, 2, 4, 7, 8, 9, 11]
pytest.raises(ValueError, lambda: is_quad_residue(1.1, 2))
pytest.raises(ValueError, lambda: is_quad_residue(2, 0))
assert quadratic_residues(12) == [0, 1, 4, 9]
assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12]
assert [len(quadratic_residues(i)) for i in range(1, 20)] == \
[1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10]
assert list(sqrt_mod_iter(6, 2)) == [0]
assert sqrt_mod(3, 13) == 4
assert sqrt_mod(3, -13) == 4
assert sqrt_mod(6, 23) == 11
assert sqrt_mod(345, 690) == 345
assert sqrt_mod(9, 18) == 3
for p in range(3, 100):
d = defaultdict(list)
for i in range(p):
d[pow(i, 2, p)].append(i)
for i in range(1, p):
it = sqrt_mod_iter(i, p)
v = sqrt_mod(i, p, True)
if v:
v = sorted(v)
assert d[i] == v
else:
assert not d[i]
assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24]
assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78]
assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240]
assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72]
assert sqrt_mod(81, 3**5, True) == [9, 18, 36, 45, 63, 72, 90, 99, 117,
126, 144, 153, 171, 180, 198, 207, 225, 234]
assert sqrt_mod(81, 3**6, True) == [9, 72, 90, 153, 171, 234, 252, 315,
333, 396, 414, 477, 495, 558, 576, 639, 657, 720]
assert sqrt_mod(81, 3**7, True) == [9, 234, 252, 477, 495, 720, 738, 963,
981, 1206, 1224, 1449, 1467, 1692, 1710, 1935, 1953, 2178]
for a, p in [(26214400, 32768000000), (26214400, 16384000000),
(262144, 1048576), (87169610025, 163443018796875),
(22315420166400, 167365651248000000)]:
assert pow(sqrt_mod(a, p), 2, p) == a
n = 70
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+2)
it = sqrt_mod_iter(a, p)
for i in range(10):
assert pow(next(it), 2, p) == a
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+3)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
n = 100
a, p = 5**2*3**n*2**n, 5**6*3**(n+1)*2**(n+1)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
assert type(next(sqrt_mod_iter(9, 27))) is int
assert isinstance(next(sqrt_mod_iter(9, 27, ZZ)), type(ZZ(1)))
assert isinstance(next(sqrt_mod_iter(1, 7, ZZ)), type(ZZ(1)))
assert list(sqrt_mod_iter(4, 919, ZZ)) == [2, 917]
assert list(sqrt_mod_iter(6, 146, ZZ)) == [88, 58]
pytest.raises(ValueError, lambda: is_nthpow_residue(+2, +1, 0))
pytest.raises(ValueError, lambda: is_nthpow_residue(+2, -1, 5))
pytest.raises(ValueError, lambda: is_nthpow_residue(-2, +1, 5))
assert is_nthpow_residue(2, 1, 5)
assert is_nthpow_residue(1, 0, 1) is False
assert is_nthpow_residue(1, 0, 2) is True
assert is_nthpow_residue(3, 0, 2) is False
assert is_nthpow_residue(0, 1, 8) is True
assert is_nthpow_residue(2, 3, 2) is False
assert is_nthpow_residue(2, 3, 9) is False
assert is_nthpow_residue(3, 5, 30) is True
assert is_nthpow_residue(21, 11, 20) is True
assert is_nthpow_residue(7, 10, 20) is False
assert is_nthpow_residue(5, 10, 20) is True
assert is_nthpow_residue(3, 10, 48) is False
assert is_nthpow_residue(1, 10, 40) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(1, 10, 24) is True
assert is_nthpow_residue(3, 10, 24) is False
assert is_nthpow_residue(2, 10, 48) is False
assert is_nthpow_residue(81, 3, 972) is False
assert is_nthpow_residue(243, 5, 5103) is True
assert is_nthpow_residue(243, 3, 1240029) is False
assert {a for a in range(1024)
if is_nthpow_residue(a, 56, 1024)} == {pow(i, 56, 1024)
for i in range(1024)}
assert {a for a in range(2048)
if is_nthpow_residue(a, 256, 2048)} == {pow(i, 256, 2048)
for i in range(2048)}
x = {pow(i, 11, 324000) for i in range(1000)}
assert [is_nthpow_residue(a, 11, 324000) for a in x]
x = {pow(i, 17, 22217575536) for i in range(1000)}
assert [is_nthpow_residue(a, 17, 22217575536) for a in x]
assert is_nthpow_residue(676, 3, 5364)
assert is_nthpow_residue(9, 12, 36)
assert is_nthpow_residue(32, 10, 41)
assert is_nthpow_residue(4, 2, 64)
assert is_nthpow_residue(31, 4, 41)
assert not is_nthpow_residue(2, 2, 5)
assert is_nthpow_residue(8547, 12, 10007)
assert nthroot_mod(1801, 11, 2663) == 44
for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663),
(26118163, 1303, 33333347), (1499, 7, 2663), (595, 6, 2663),
(1714, 12, 2663), (28477, 9, 33343)]:
r = nthroot_mod(a, q, p)
assert pow(r, q, p) == a
assert nthroot_mod(11, 3, 109) is None
assert nthroot_mod(6, 12, 5) == 1
pytest.raises(NotImplementedError, lambda: nthroot_mod(16, 5, 36))
pytest.raises(NotImplementedError, lambda: nthroot_mod(9, 16, 36))
for p in primerange(5, 100):
qv = range(3, p, 4)
for q in qv:
d = defaultdict(list)
for i in range(p):
d[pow(i, q, p)].append(i)
for a in range(1, p - 1):
res = nthroot_mod(a, q, p, True)
if d[a]:
assert d[a] == res
else:
assert res is None
assert legendre_symbol(5, 11) == 1
assert legendre_symbol(25, 41) == 1
assert legendre_symbol(67, 101) == -1
assert legendre_symbol(0, 13) == 0
assert legendre_symbol(9, 3) == 0
pytest.raises(ValueError, lambda: legendre_symbol(2, 4))
assert jacobi_symbol(25, 41) == 1
assert jacobi_symbol(-23, 83) == -1
assert jacobi_symbol(3, 9) == 0
assert jacobi_symbol(42, 97) == -1
assert jacobi_symbol(3, 5) == -1
assert jacobi_symbol(7, 9) == 1
assert jacobi_symbol(0, 3) == 0
assert jacobi_symbol(0, 1) == 1
assert jacobi_symbol(2, 1) == 1
assert jacobi_symbol(1, 3) == 1
pytest.raises(ValueError, lambda: jacobi_symbol(3, 8))
assert mobius(13*7) == 1
assert mobius(1) == 1
assert mobius(13*7*5) == -1
assert mobius(13**2) == 0
pytest.raises(ValueError, lambda: mobius(-3))
p = Symbol('p', integer=True, positive=True, prime=True)
x = Symbol('x', positive=True)
i = Symbol('i', integer=True)
assert mobius(p) == -1
pytest.raises(TypeError, lambda: mobius(x))
pytest.raises(ValueError, lambda: mobius(i))
mobius(Symbol('p', positive=True, integer=True))
def test_residue():
assert n_order(2, 13) == 12
assert [n_order(a, 7) for a in range(1, 7)] == \
[1, 3, 6, 3, 6, 2]
assert n_order(5, 17) == 16
assert n_order(17, 11) == n_order(6, 11)
assert n_order(101, 119) == 6
assert n_order(
11, (10**50 + 151)**2
) == 10000000000000000000000000000000000000000000000030100000000000000000000000000000000000000000000022650
pytest.raises(ValueError, lambda: n_order(6, 9))
assert is_primitive_root(2, 7) is False
assert is_primitive_root(3, 8) is False
assert is_primitive_root(11, 14) is False
assert is_primitive_root(12, 17) == is_primitive_root(29, 17)
pytest.raises(ValueError, lambda: is_primitive_root(3, 6))
assert [primitive_root(i) for i in range(2, 31)] == [
1, 2, 3, 2, 5, 3, None, 2, 3, 2, None, 2, 3, None, None, 3, 5, 2, None,
None, 7, 5, None, 2, 7, 2, None, 2, None
]
for p in primerange(3, 100):
it = _primitive_root_prime_iter(p)
assert len(list(it)) == totient(totient(p))
assert primitive_root(97) == 5
assert primitive_root(97**2) == 5
assert primitive_root(40487) == 5
# note that primitive_root(40487) + 40487 = 40492 is a primitive root
# of 40487**2, but it is not the smallest
assert primitive_root(40487**2) == 10
assert primitive_root(82) == 7
p = 10**50 + 151
assert primitive_root(p) == 11
assert primitive_root(2 * p) == 11
assert primitive_root(p**2) == 11
pytest.raises(ValueError, lambda: primitive_root(-3))
assert is_quad_residue(3, 7) is False
assert is_quad_residue(10, 13) is True
assert is_quad_residue(12364, 139) == is_quad_residue(12364 % 139, 139)
assert is_quad_residue(207, 251) is True
assert is_quad_residue(0, 1) is True
assert is_quad_residue(1, 1) is True
assert is_quad_residue(0, 2) == is_quad_residue(1, 2) is True
assert is_quad_residue(1, 4) is True
assert is_quad_residue(2, 27) is False
assert is_quad_residue(13122380800, 13604889600) is True
assert [j for j in range(14) if is_quad_residue(j, 14)] == \
[0, 1, 2, 4, 7, 8, 9, 11]
pytest.raises(ValueError, lambda: is_quad_residue(1.1, 2))
pytest.raises(ValueError, lambda: is_quad_residue(2, 0))
assert quadratic_residues(12) == [0, 1, 4, 9]
assert quadratic_residues(13) == [0, 1, 3, 4, 9, 10, 12]
assert [len(quadratic_residues(i)) for i in range(1, 20)] == \
[1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10]
assert list(sqrt_mod_iter(6, 2)) == [0]
assert sqrt_mod(3, 13) == 4
assert sqrt_mod(3, -13) == 4
assert sqrt_mod(6, 23) == 11
assert sqrt_mod(345, 690) == 345
assert sqrt_mod(9, 18) == 3
for p in range(3, 100):
d = defaultdict(list)
for i in range(p):
d[pow(i, 2, p)].append(i)
for i in range(1, p):
it = sqrt_mod_iter(i, p)
v = sqrt_mod(i, p, True)
if v:
v = sorted(v)
assert d[i] == v
else:
assert not d[i]
assert sqrt_mod(9, 27, True) == [3, 6, 12, 15, 21, 24]
assert sqrt_mod(9, 81, True) == [3, 24, 30, 51, 57, 78]
assert sqrt_mod(9, 3**5, True) == [3, 78, 84, 159, 165, 240]
assert sqrt_mod(81, 3**4, True) == [0, 9, 18, 27, 36, 45, 54, 63, 72]
assert sqrt_mod(81, 3**5, True) == [
9, 18, 36, 45, 63, 72, 90, 99, 117, 126, 144, 153, 171, 180, 198, 207,
225, 234
]
assert sqrt_mod(81, 3**6, True) == [
9, 72, 90, 153, 171, 234, 252, 315, 333, 396, 414, 477, 495, 558, 576,
639, 657, 720
]
assert sqrt_mod(81, 3**7, True) == [
9, 234, 252, 477, 495, 720, 738, 963, 981, 1206, 1224, 1449, 1467,
1692, 1710, 1935, 1953, 2178
]
for a, p in [(26214400, 32768000000), (26214400, 16384000000),
(262144, 1048576), (87169610025, 163443018796875),
(22315420166400, 167365651248000000)]:
assert pow(sqrt_mod(a, p), 2, p) == a
n = 70
a, p = 5**2 * 3**n * 2**n, 5**6 * 3**(n + 1) * 2**(n + 2)
it = sqrt_mod_iter(a, p)
for i in range(10):
assert pow(next(it), 2, p) == a
a, p = 5**2 * 3**n * 2**n, 5**6 * 3**(n + 1) * 2**(n + 3)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
n = 100
a, p = 5**2 * 3**n * 2**n, 5**6 * 3**(n + 1) * 2**(n + 1)
it = sqrt_mod_iter(a, p)
for i in range(2):
assert pow(next(it), 2, p) == a
assert type(next(sqrt_mod_iter(9, 27))) is int
assert isinstance(next(sqrt_mod_iter(9, 27, ZZ)), type(ZZ(1)))
assert isinstance(next(sqrt_mod_iter(1, 7, ZZ)), type(ZZ(1)))
assert list(sqrt_mod_iter(4, 919, ZZ)) == [2, 917]
assert list(sqrt_mod_iter(6, 146, ZZ)) == [88, 58]
assert is_nthpow_residue(2, 1, 5)
assert not is_nthpow_residue(2, 2, 5)
assert is_nthpow_residue(8547, 12, 10007)
assert nthroot_mod(1801, 11, 2663) == 44
for a, q, p in [(51922, 2, 203017), (43, 3, 109), (1801, 11, 2663),
(26118163, 1303, 33333347), (1499, 7, 2663),
(595, 6, 2663), (1714, 12, 2663), (28477, 9, 33343)]:
r = nthroot_mod(a, q, p)
assert pow(r, q, p) == a
assert nthroot_mod(11, 3, 109) is None
assert nthroot_mod(6, 12, 5) == 1
for p in primerange(5, 100):
qv = range(3, p, 4)
for q in qv:
d = defaultdict(list)
for i in range(p):
d[pow(i, q, p)].append(i)
for a in range(1, p - 1):
res = nthroot_mod(a, q, p, True)
if d[a]:
assert d[a] == res
else:
assert res is None
assert legendre_symbol(5, 11) == 1
assert legendre_symbol(25, 41) == 1
assert legendre_symbol(67, 101) == -1
assert legendre_symbol(0, 13) == 0
assert legendre_symbol(9, 3) == 0
pytest.raises(ValueError, lambda: legendre_symbol(2, 4))
assert jacobi_symbol(25, 41) == 1
assert jacobi_symbol(-23, 83) == -1
assert jacobi_symbol(3, 9) == 0
assert jacobi_symbol(42, 97) == -1
assert jacobi_symbol(3, 5) == -1
assert jacobi_symbol(7, 9) == 1
assert jacobi_symbol(0, 3) == 0
assert jacobi_symbol(0, 1) == 1
assert jacobi_symbol(2, 1) == 1
assert jacobi_symbol(1, 3) == 1
pytest.raises(ValueError, lambda: jacobi_symbol(3, 8))
assert mobius(13 * 7) == 1
assert mobius(1) == 1
assert mobius(13 * 7 * 5) == -1
assert mobius(13**2) == 0
pytest.raises(ValueError, lambda: mobius(-3))
p = Symbol('p', integer=True, positive=True, prime=True)
x = Symbol('x', positive=True)
i = Symbol('i', integer=True)
assert mobius(p) == -1
pytest.raises(TypeError, lambda: mobius(x))
pytest.raises(ValueError, lambda: mobius(i))
mobius(Symbol('p', positive=True, integer=True))
