Python ntheory.primerange函数代码示例

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)
    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(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

    raises(ValueError, lambda: totient(0))

    raises(ValueError, lambda: reduced_totient(0))

    raises(ValueError, lambda: primorial(0))

    assert mr(1, [2]) is False

    func = lambda i: (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]

def test_primeomega():
    assert primeomega(2) == 1
    assert primeomega(2 * 2) == 2
    assert primeomega(2 * 2 * 3) == 3
    assert primeomega(3 * 25) == primeomega(3) + primeomega(25)
    assert [primeomega(p) for p in primerange(1, 10)] == [1, 1, 1, 1]
    assert primeomega(fac(50)) == 108
    assert primeomega(2 ** 9941 - 1) == 1
    n = Symbol('n', integer=True)
    assert primeomega(n)
    assert primeomega(n).subs(n, 2 ** 31 - 1) == 1
    assert summation(primeomega(n), (n, 2, 30)) == 59

def sumcons(Nmin, Nmax):
    retval = (-1, -1)
    tot = 0
    for i, n in enumerate(nt.primerange(Nmin, Nmax)):
        if i == 0:
            continue
        if tot >= Nmax:
            break
        if nt.isprime(tot):
            retval = (i, tot)
        tot = tot + n
    return retval

def test_generate():
    assert nextprime(-4) == 2
    assert nextprime(2) == 3
    assert nextprime(5) == 7
    assert nextprime(90) == 97
    assert nextprime(10**40) == (10**40 + 121)
    assert prevprime(3) == 2
    assert prevprime(7) == 5
    assert prevprime(97) == 89
    assert prevprime(10**40) == (10**40 - 17)
    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

def count_prime_power_triples(threshold):
    primes = [p for p in ntheory.primerange(1, math.sqrt(threshold))]
    numbers = set()
    for fourth_root in primes:
        n4 = fourth_root ** 4
        if n4 >= threshold:
            break
        for cube_root in primes:
            n3 = cube_root ** 3
            if n4 + n3 >= threshold:
                break
            for square_root in primes:
                n2 = square_root ** 2
                s = n4 + n3 + n2
                if s >= threshold:
                    break
                # print "%d^2 + %d^3 + %d^4 = %d" % (square_root, cube_root, fourth_root, s)
                numbers.add(s)
    return len(numbers)

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
    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)
    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
    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]
    raises(ValueError, lambda: is_quad_residue(1.1, 2))
    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

    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):
#.........这里部分代码省略.........

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
    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)
    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
    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]
    raises(ValueError, lambda: is_quad_residue(1.1, 2))
    raises(ValueError, lambda: is_quad_residue(2, 0))


    assert quadratic_residues(S.One) == [0]
    assert quadratic_residues(1) == [0]
    assert quadratic_residues(12) == [0, 1, 4, 9]
    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

    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
#.........这里部分代码省略.........

def test_generate():
    from sympy.ntheory.generate import sieve
    sieve._reset()
    assert nextprime(-4) == 2
    assert nextprime(2) == 3
    assert nextprime(5) == 7
    assert nextprime(12) == 13
    assert prevprime(3) == 2
    assert prevprime(7) == 5
    assert prevprime(13) == 11
    assert prevprime(19) == 17
    assert prevprime(20) == 19

    sieve.extend_to_no(9)
    assert sieve._list[-1] == 23

    assert sieve._list[-1] < 31
    assert 31 in sieve

    assert nextprime(90) == 97
    assert nextprime(10**40) == (10**40 + 121)
    assert prevprime(97) == 89
    assert prevprime(10**40) == (10**40 - 17)

    assert list(sieve.primerange(10, 1)) == []
    assert list(sieve.primerange(5, 9)) == [5, 7]
    sieve._reset(prime=True)
    assert list(sieve.primerange(2, 12)) == [2, 3, 5, 7, 11]

    assert list(sieve.totientrange(5, 15)) == [4, 2, 6, 4, 6, 4, 10, 4, 12, 6]
    sieve._reset(totient=True)
    assert list(sieve.totientrange(3, 13)) == [2, 2, 4, 2, 6, 4, 6, 4, 10, 4]
    assert list(sieve.totientrange(900, 1000)) == [totient(x) for x in range(900, 1000)]
    assert list(sieve.totientrange(0, 1)) == []
    assert list(sieve.totientrange(1, 2)) == [1]

    assert list(sieve.mobiusrange(5, 15)) == [-1, 1, -1, 0, 0, 1, -1, 0, -1, 1]
    sieve._reset(mobius=True)
    assert list(sieve.mobiusrange(3, 13)) == [-1, 0, -1, 1, -1, 0, 0, 1, -1, 0]
    assert list(sieve.mobiusrange(1050, 1100)) == [mobius(x) for x in range(1050, 1100)]
    assert list(sieve.mobiusrange(0, 1)) == []
    assert list(sieve.mobiusrange(1, 2)) == [1]

    assert list(primerange(10, 1)) == []
    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

    raises(ValueError, lambda: totient(0))

    raises(ValueError, lambda: reduced_totient(0))

    raises(ValueError, lambda: primorial(0))

    assert mr(1, [2]) is False

    func = lambda i: (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]
    sieve.extend(3000)
    assert nextprime(2968) == 2969
    assert prevprime(2930) == 2927
    raises(ValueError, lambda: prevprime(1))