def test_prime_generator():
prime_generator_limit_cases = [
(60, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59]),
(1, [])
]
prime_generator_count_cases = [
(10, [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]),
(1, [2]),
[0, []]
]
for inp, output in prime_generator_limit_cases:
assert list(prime_generator(limit=inp)) == output
for inp, output in prime_generator_count_cases:
assert list(prime_generator(count=inp)) == output
def compute():
LIMIT = 10**8
MOD = 1000000009
ans = 1
for p in eulerlib.prime_generator(LIMIT):
power = count_factors(LIMIT, p)
ans *= 1 + pow(p, power * 2, MOD)
ans %= MOD
return str(ans)
def compute():
LIMIT = 10 ** 8
MOD = 1000000009
ans = 1
for p in eulerlib.prime_generator(LIMIT):
power = count_factors(LIMIT, p)
ans *= 1 + pow(p, power * 2, MOD)
ans %= MOD
return str(ans)
def main():
total = 100000000
primes_set = set(tqdm(prime_generator(limit=total), total=total))
s = 0
for i in tqdm(primes_set):
flag = True
for d in proper_divisors(i - 1):
if d + (i - 1) // d not in primes_set:
flag = False
break
if flag:
s += i - 1
print(s)
def compute():
# Note about the mathematical simplification:
# (p-5)! + (p-4)! + (p-3)! + (p-2)! + (p-1)!
# = (p-5)! * (1 + (p-4) + (p-4)(p-3) + (p-4)(p-3)(p-2) + (p-4)(p-3)(p-2)(p-1))
# = (p-5)! * (1 + (-4) + (-4)(-3) + (-4)(-3)(-2) + (-4)(-3)(-2)(-1))
# = (p-5)! * (1 + -4 + 12 + -24 + 24)
# = (p-5)! * 9
# = (p-1)! / ((p-1)(p-2)(p-3)(p-4)) * 9
# = (p-1)! / ((-1)(-2)(-3)(-4)) * 9
# = (p-1)! / 24 * 9
# = (p-1)! * (3 * 3) / (3 * 8)
# = (p-1)! * 3 / 8
# = -1 * 3 / 8 (by Wilson's theorem)
# = -3/8 mod p.
# Every part of the equation is modulo a prime p > 4.
def s(p):
return (p - 3) * eulerlib.reciprocal_mod(8 % p, p) % p
ans = sum(s(p) for p in eulerlib.prime_generator(10**8) if p >= 5)
return str(ans)
def compute(): # Note about the mathematical simplification: # (p-5)! + (p-4)! + (p-3)! + (p-2)! + (p-1)! # = (p-5)! * (1 + (p-4) + (p-4)(p-3) + (p-4)(p-3)(p-2) + (p-4)(p-3)(p-2)(p-1)) # = (p-5)! * (1 + (-4) + (-4)(-3) + (-4)(-3)(-2) + (-4)(-3)(-2)(-1)) # = (p-5)! * (1 + -4 + 12 + -24 + 24) # = (p-5)! * 9 # = (p-1)! / ((p-1)(p-2)(p-3)(p-4)) * 9 # = (p-1)! / ((-1)(-2)(-3)(-4)) * 9 # = (p-1)! / 24 * 9 # = (p-1)! * (3 * 3) / (3 * 8) # = (p-1)! * 3 / 8 # = -1 * 3 / 8 (by Wilson's theorem) # = -3/8 mod p. # Every part of the equation is modulo a prime p > 4. def s(p): return (p - 3) * reciprocal_mod(8 % p, p) % p ans = sum(s(p) for p in eulerlib.prime_generator(10**8) if p >= 5) return str(ans)
import eulerlib
import math
import time
once = time.time()
pliste = eulerlib.prime_generator(10**8)
print(pliste(13))
def is_prime(a):
if x <= 1:
return False
elif x <= 3:
return True
elif x % 2 == 0:
return False
else:
for i in range(3, sqrt(x) + 1, 2):
if x % i == 0:
return False
return True
def func(p):
for i in range(1, p):
if (8 * i + 3) % p == 0:
return i
toplam = 0
for p in pliste:
toplam = toplam + func(p)