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(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))
def S(N): s = 0 for i in primerange(2, N ** 0.5 + 1): for j in primerange(i + 1, N // 2): prod = i * j if prod > N: break maxProd = 0 while prod <= N: curr = prod while curr * j <= N: curr *= j if maxProd < curr: maxProd = curr prod *= i s += maxProd return s
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 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_primenu(): assert primenu(2) == 1 assert primenu(2 * 3) == 2 assert primenu(2 * 3 * 5) == 3 assert primenu(3 * 25) == primenu(3) + primenu(25) assert [primenu(p) for p in primerange(1, 10)] == [1, 1, 1, 1] assert primenu(fac(50)) == 15 assert primenu(2**9941 - 1) == 1 n = Symbol("n", integer=True) assert primenu(n) assert primenu(n).subs(n, 2**31 - 1) == 1 assert summation(primenu(n), (n, 2, 30)) == 43
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 prime_numbers(self, input_a, input_b): prime_numbers_arr = [i for i in primerange(input_a, input_b+1)] # current_number = input_a # prime_numbers_arr = [] # while current_number <= input_b: # test_result = self.if_prime(current_number) # if test_result: # prime_numbers_arr.append(current_number) # current_number += 1 return prime_numbers_arr
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 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 euler399(n): print("generating Fibonacci sieve") limit = int(n * 1.31) sffs = bitarray(limit) sffs.setall(True) # square free Fibonacci sieve for p in primerange(1, 1500000): l = p * fiblen(p, limit // p + 1) if l: sffs[l - 1::l] = False print("counting") t0 = time() i = count_n(sffs, n) print("time:", time() - t0) print(i) print('%i,%.1fe%i' % ith_fib(i))
def p659(kmax): t0 = time() primes = list(primerange(1, 2 * kmax)) primes = [prime for prime in primes if (prime - 1) % 4 == 0] total = 0 for k in range(1, kmax + 1): rem = 4 * k * k + 1 biggest_prime_divisor = 1 for prime in primes: while prime * prime <= rem and rem % prime == 0: rem //= prime biggest_prime_divisor = prime if prime * prime > rem: break total += biggest_prime_divisor if rem == 1 else rem return total, time() - t0
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 totient(n): q = 2 a = 0 i = 0 p = list(primerange(2, int(1.2 * sqrt(n)))) del p[:int(0.6 * len(p))] for j in p: i += 1 for k in p[i:]: if (j + k) % 9 != 1: continue b = j * k if b > n: return (a) t = (j - 1) * (k - 1) c = b / float(t) if permutation(t, b) and q > c: q = c a = b
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)) 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 assert sqrt_mod(67, 101) == None assert sqrt_mod(1020, 104729) == None 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 type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) assert is_nthpow_residue(2, 1, 5) #issue 10816 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 True 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 is_nthpow_residue(36010, 8, 87382) is True assert is_nthpow_residue(28552, 6, 2218) is True assert is_nthpow_residue(92712, 9, 50026) is True x = set([pow(i, 56, 1024) for i in range(1024)]) assert set([a for a in range(1024) if is_nthpow_residue(a, 56, 1024)]) == x x = set([pow(i, 256, 2048) for i in range(2048)]) assert set([a for a in range(2048) if is_nthpow_residue(a, 256, 2048)]) == x x = set([pow(i, 11, 324000) for i in range(1000)]) assert [is_nthpow_residue(a, 11, 324000) for a in x] x = set([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(29, 31, 74) == [45] 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(16, 5, 36, True) == [4, 22] assert nthroot_mod(9, 16, 36, True) == [3, 9, 15, 21, 27, 33] assert nthroot_mod(4, 3, 3249000) == [] assert nthroot_mod(36010, 8, 87382, True) == [40208, 47174] assert nthroot_mod(0, 12, 37, True) == [0] assert nthroot_mod(0, 7, 100, True) == [0, 10, 20, 30, 40, 50, 60, 70, 80, 90] assert nthroot_mod(4, 4, 27, True) == [5, 22] assert nthroot_mod(4, 4, 121, True) == [19, 102] assert nthroot_mod(2, 3, 7, True) == [] for p in range(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 == [] 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 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 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 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 raises(TypeError, lambda: mobius(x)) raises(ValueError, lambda: mobius(i)) 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 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 assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6 assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19 assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40 assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333 raises(ValueError, lambda: _discrete_log_pollard_rho(11, 7, 31, rseed=0)) raises(ValueError, lambda: _discrete_log_pollard_rho(227, 3**7, 5, rseed=0)) 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 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 args = 5779, 3528, 6215 assert discrete_log(*args) == 687 assert discrete_log(*Tuple(*args)) == 687 assert quadratic_congruence(400, 85, 125, 1600) == [295, 615, 935, 1255, 1575] assert quadratic_congruence(3, 6, 5, 25) == [3, 20] assert quadratic_congruence(120, 80, 175, 500) == [] assert quadratic_congruence(15, 14, 7, 2) == [1] assert quadratic_congruence(8, 15, 7, 29) == [10, 28] assert quadratic_congruence(160, 200, 300, 461) == [144, 431] assert quadratic_congruence( 100000, 123456, 7415263, 48112959837082048697) == [30417843635344493501, 36001135160550533083] assert quadratic_congruence(65, 121, 72, 277) == [249, 252] assert quadratic_congruence(5, 10, 14, 2) == [0] assert quadratic_congruence(10, 17, 19, 2) == [1] assert quadratic_congruence(10, 14, 20, 2) == [0, 1] assert polynomial_congruence( 6 * x**5 + 10 * x**4 + 5 * x**3 + x**2 + x + 1, 972000) == [ 220999, 242999, 463999, 485999, 706999, 728999, 949999, 971999 ] assert polynomial_congruence(x**3 - 10 * x**2 + 12 * x - 82, 33075) == [30287] assert polynomial_congruence(x**2 + x + 47, 2401) == [785, 1615] assert polynomial_congruence(10 * x**2 + 14 * x + 20, 2) == [0, 1] assert polynomial_congruence(x**3 + 3, 16) == [5] assert polynomial_congruence(65 * x**2 + 121 * x + 72, 277) == [249, 252] assert polynomial_congruence(35 * x**3 - 6 * x**2 - 567 * x + 2308, 148225) == [86957, 111157, 122531, 146731] assert polynomial_congruence(x**16 - 9, 36) == [3, 9, 15, 21, 27, 33] assert polynomial_congruence(x**6 - 2 * x**5 - 35, 6125) == [3257] raises(ValueError, lambda: polynomial_congruence(x**x, 6125)) raises(ValueError, lambda: polynomial_congruence(x**i, 6125)) raises(ValueError, lambda: polynomial_congruence(0.1 * x**2 + 6, 100))
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): 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 type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) 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 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 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 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 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 raises(TypeError, lambda: mobius(x)) raises(ValueError, lambda: mobius(i))
import itertools from functools import lru_cache from sympy.ntheory import primerange # this is basically the same as problem 31 coins = list(primerange(1, 100)) Ncoins = len(coins) @lru_cache(1024) def count(amount, maxcoin=0): res = 0 for i in range(maxcoin, Ncoins): amount_left = amount - coins[i] if amount_left == 0: res += 1 if amount_left > 0: res += count(amount_left, i) return res for n in itertools.count(2): if count(n) > 5000: print(n) break
def if_prime(self, number): check_result = [i for i in primerange(number, number + 1)] return len(check_result) == 1
from bisect import bisect_left from math import log, prod from sympy.ntheory import primerange LIMIT = 190 primes = list(primerange(1, LIMIT)) log_primes = [log(x) for x in primes] NP = len(primes) print('NP', NP) # I did not think about what happens in case the number of primes is odd assert NP % 2 == 0 NH = NP // 2 p = prod(primes) print('p', p) hlogp = log(p) / 2 print('log(p)/2', hlogp) NB = 1 << NH lower = [] upper = [] for k in range(NB): lsum = usum = 0 for i in range(NH): if (1 << i) & k: lsum += log_primes[i] usum += log_primes[i + NH] lower.append((lsum, k)) upper.append((usum, k))
# very slow about 10 minutes from sympy.ntheory import totient, primerange CHAIN_LEN = 25 PRIME_LIMIT = 40_000_000 cache = {} def chain(n): n -= 1 # if n is prime, then totient(n) = n - 1 m = n s = 0 while n > 1: if n in cache: s += cache[n] break n = totient(n) s += 1 cache[m] = s return s + 2 == CHAIN_LEN result = 0 for p in primerange(1, PRIME_LIMIT): if chain(p): result += p print(len(cache)) print(result)
from sympy import mod_inverse from sympy.ntheory import primerange, isprime def facmod(n, m): if n >= m: return 0 res = m - 1 for i in range(m - 1, n, -1): res *= mod_inverse(i, m) res %= m return res def S(i): assert isprime(i) minus5 = facmod(i - 5, i) minus4 = (minus5 * (i - 4)) % i minus3 = (minus4 * (i - 3)) % i minus2 = (minus3 * (i - 2)) % i minus1 = i - 1 # Wilson's theorem return (minus1 + minus2 + minus3 + minus4 + minus5) % i res = 0 for n, p in enumerate(primerange(5, 1E8)): res += S(p) if n % 1000 == 0: print(p, res) print(res)
from fractions import Fraction from sympy.ntheory import totient, primerange def R(d): return Fraction(totient(d), d - 1) TH = Fraction(15499, 94744) d = 1 for p in primerange(1, 100): d *= p r = R(d) print(d, p, r) if r < TH: break pp = d // p print(pp) for k in range(1, p): d = k * pp r = R(d) print(d, k, r) if r < TH: break print(d)
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): 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 type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) assert is_nthpow_residue(2, 1, 5) #issue 10816 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 x = set([pow(i, 56, 1024) for i in range(1024)]) assert set([a for a in range(1024) if is_nthpow_residue(a, 56, 1024)]) == x x = set([pow(i, 256, 2048) for i in range(2048)]) assert set([a for a in range(2048) if is_nthpow_residue(a, 256, 2048)]) == x x = set([pow(i, 11, 324000) for i in range(1000)]) assert [is_nthpow_residue(a, 11, 324000) for a in x] x = set([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 raises(NotImplementedError, lambda: nthroot_mod(16, 5, 36)) 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 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 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 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 raises(TypeError, lambda: mobius(x)) raises(ValueError, lambda: mobius(i)) 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 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 assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6 assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19 assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40 assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333 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 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
def primorial_totient(n): """Returns the totient of a given primorial.""" return reduce(mul, (p - 1 for p in nthry.primerange(2, nthry.prime(n + 1))))
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 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 type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) assert is_nthpow_residue(2, 1, 5) #issue 10816 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 x = set([pow(i, 56, 1024) for i in range(1024)]) assert set([a for a in range(1024) if is_nthpow_residue(a, 56, 1024)]) == x x = set([ pow(i, 256, 2048) for i in range(2048)]) assert set([a for a in range(2048) if is_nthpow_residue(a, 256, 2048)]) == x x = set([ pow(i, 11, 324000) for i in range(1000)]) assert [ is_nthpow_residue(a, 11, 324000) for a in x] x = set([ 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(29, 31, 74) == 31 assert nthroot_mod(*Tuple(29, 31, 74)) == 31 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 raises(NotImplementedError, lambda: nthroot_mod(16, 5, 36)) 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 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 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 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 raises(TypeError, lambda: mobius(x)) raises(ValueError, lambda: mobius(i)) 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 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 assert _discrete_log_pollard_rho(6013199, 2**6, 2, rseed=0) == 6 assert _discrete_log_pollard_rho(6138719, 2**19, 2, rseed=0) == 19 assert _discrete_log_pollard_rho(36721943, 2**40, 2, rseed=0) == 40 assert _discrete_log_pollard_rho(24567899, 3**333, 3, rseed=0) == 333 raises(ValueError, lambda: _discrete_log_pollard_rho(11, 7, 31, rseed=0)) raises(ValueError, lambda: _discrete_log_pollard_rho(227, 3**7, 5, rseed=0)) 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 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 args = 5779, 3528, 6215 assert discrete_log(*args) == 687 assert discrete_log(*Tuple(*args)) == 687
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))
from array import array from sympy.ntheory import primerange n = 50000 MOD = 1000_000_000 f = array('i', n * [0]) f[0] = 1 for p in primerange(1, n): for i in range(p, n): f[i] = (f[i] + f[i - p] * p) % MOD print(p) for k in range(20): print(k, f[k]) ans = 0 a = b = 1 for k in range(2, 25): c = a + b print('a=', a) ans = (ans + f[a]) % MOD b, a = a, c print(ans)
from math import ceil, sqrt from bitarray import bitarray from sympy.ntheory import primerange LIMIT = 50_000_000 primes = list(primerange(1, ceil(sqrt(LIMIT)))) ba = bitarray(LIMIT) ba.setall(0) for a in primes: a2 = pow(a, 2) for b in primes: b3 = pow(b, 3) if b3 > LIMIT: break for c in primes: n = a2 + b3 + pow(c, 4) if n >= LIMIT: break ba[n] = 1 print(ba.count())
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): 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 type(next(sqrt_mod_iter(9, 27, ZZ))) is type(ZZ(1)) assert type(next(sqrt_mod_iter(1, 7, ZZ))) is type(ZZ(1)) 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 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 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 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 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 raises(TypeError, lambda: mobius(x)) raises(ValueError, lambda: mobius(i))
from sympy.ntheory import primerange, isprime N = 1_000_000 primes = list(primerange(1, N)) ls = 0 for s in range(len(primes) - 2000): r = primes[s] for t in range(1, 2000): r += primes[s + t] if r > N: break if isprime(r): if t + 1 > ls: print(r, t + 1) ls = t + 1
def remainder(n, l): for i, j in enumerate(primerange(1, 10**l)): if (((j - 1)**(i + 1)) + ((j + 1)**(i + 1))) % (j**2) > 10**n: print(i + 1) break