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 prime_generators():
def simple_primes():
D = defaultdict(list)
for q in naturals(2):
if q not in D:
yield q
D[q * q] = [q]
else:
for p in D[q]:
D[p + q].append(p)
del D[q]
def erat30():
"https://stackoverflow.com/questions/2211990/how-to-implement-an-efficient-infinite-generator-of-prime-numbers-in-python"
D = {9: 3, 25: 5}
yield 2
yield 3
yield 5
MASK = cycle((1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0))
MODULOS = frozenset((1, 7, 11, 13, 17, 19, 23, 29))
for q in compress(count(7, 2), MASK):
p = D.pop(q, None)
if p is None:
D[q * q] = q
yield q
else:
x = q + 2 * p
while x in D or (x % 30) not in MODULOS:
x += 2 * p
D[x] = p
sieve._reset()
def primes_sympy():
upper = 2
while True:
upper *= 2
yield from sieve.primerange(upper // 2, upper)
def primes_sympy2():
upper = 2
while True:
upper *= 2
yield from sieve.primerange(upper // 2, upper)
print("Generate the first 40000 prime numbers")
speed_compare(
[simple_primes(),
erat30(), primes_sympy(),
primes_sympy2()], ["Simple", "erat30", "sympy", "sympy_again"],
n=40000,
reps=3)
def test_composite():
from sympy.ntheory.generate import sieve
sieve._reset()
assert composite(1) == 4
assert composite(2) == 6
assert composite(5) == 10
assert composite(11) == 20
assert composite(41) == 58
assert composite(57) == 80
assert composite(296) == 370
assert composite(559) == 684
assert composite(3000) == 3488
assert composite(4096) == 4736
assert composite(9096) == 10368
assert composite(25023) == 28088
sieve.extend(3000)
assert composite(1957) == 2300
assert composite(2568) == 2998
raises(ValueError, lambda: composite(0))
def test_composite():
from sympy.ntheory.generate import sieve
sieve._reset()
assert composite(1) == 4
assert composite(2) == 6
assert composite(5) == 10
assert composite(11) == 20
assert composite(41) == 58
assert composite(57) == 80
assert composite(296) == 370
assert composite(559) == 684
assert composite(3000) == 3488
assert composite(4096) == 4736
assert composite(9096) == 10368
assert composite(25023) == 28088
sieve.extend(3000)
assert composite(1957) == 2300
assert composite(2568) == 2998
raises(ValueError, lambda: composite(0))
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))
#NumberTh functions with SymPy package
#-------------------------------------
from sympy import ntheory
from sympy import sieve
from sympy import generate
sieve._reset(
) # An infinite list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes.#When a lookup is
# requested involving an odd number that has not been sieved,the sieve is automatically extended up to that number.
print(13 in sieve) # prints 'True' if the input is prime,
sieve.extend(40) #for input n grows the sieve to contain all primes <= n
print(sieve[10]) # returns n th prime
sieve.extend_to_no(15) # extends upto n th prime
# mobiusrange(a,b) genarates Mobius numbers for the range [a,b), means output will be a list.
Mob_func = sieve.mobiusrange(7, 12)
print('\n Mobius function outputs in the given range:', [i for i in Mob_func])
# primerange(a,b) generates all prime numbers in the range [a,b)
Prime_list = sieve.primerange(1, 5)
print('\n Primes in the given range are :', [i for i in Prime_list])
#search(n) return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression,
# if ceiling cannot convert it to an integer then an error will be raised.
x, y = sieve.search(25)
print('\n The given input is in between ', '(', x, ',', y, ')', 'th Primes \n')
#used to check which values exist
existence = multiprocessing.Array('i', [0 for val in range(max_n + 1)])
existence[1] = 1
existence[2] = 1
existence[3] = 1
#holds our processes
jobs = []
#prime sieve to generate primes
#we use these for finding the smallest prime divisor of each n
print("Calculating all necessary primes...")
primes = multiprocessing.Array(
'i', [i for i in sieve.primerange(2,
int(sqrt(max_n)) + 1)])
sieve._reset()
print("Solving...")
##
#Creating the processes
##
#setting up each process with their own values of n
#each of the k processes only looks at values of n where n+cur=0(mod k), 0<cur<k
#initialize the processes in a loop
#pass in the shared variables and start the processing
for cur in range(0, processes):
cur_process = multiprocessing.Process(
target=inductive_solver,
