def main():
primes = prime_under(10**4)
primes.remove(5)
p1, p2, p3, p4, p5 = find_solution5(primes)
print(p1, p2, p3, p4, p5)
print(sum([p1, p2, p3, p4, p5]))
def main():
primes = prime_under(10**4)
primes.remove(5)
p1, p2, p3, p4, p5 = find_solution5(primes)
print(p1, p2, p3, p4, p5)
print(sum([p1, p2, p3, p4, p5]))
def main():
first40 = []
count = 0
for p in prime_under(10 ** 6)[3:]: # exclude 2 and 5, which are not coprime with 10
if repunit_mod(p) == 0:
count += 1
# print(count, p)
first40.append(p)
if count == 40:
break
# print(first40)
print(sum(first40))
def main():
first40 = []
count = 0
for p in prime_under(
10**6)[3:]: # exclude 2 and 5, which are not coprime with 10
if repunit_mod(p) == 0:
count += 1
# print(count, p)
first40.append(p)
if count == 40:
break
# print(first40)
print(sum(first40))
def decomp_matrix(n):
primes = prime_under(n + 1)
primes_exp = [p**(int(log(n, p)) * 2) for p in primes]
mat = [[0] * (n - 1) for _ in range(len(primes))]
for i in range(2, n + 1):
decomp = decompose(i)
for coeff, fact_exp, fact in decomp:
idx = primes.index(fact)
mat[idx][i - 2] = coeff * (primes_exp[idx] // fact_exp)
return mat, primes_exp
def main():
ceiling = 5 * 10**7
primes = prime_under(int((ceiling - 2**3 - 2**4)**0.5) + 1)
nums = set()
print("Primes generated by:", time() - starting_time, "seconds")
for i in primes:
for j in primes:
for k in primes:
num = i**4 + j**3 + k**2
if num > ceiling:
break
nums.add(num)
print(len(nums))
def main():
N = 4 * 10**6
N2 = N * 2 + 1
primes = prime_under(50)
# print(log(N2, 3), len(primes), list(enumerate(primes)), sep='\n')
choice = []
for i3 in range(ceil(log(N2, 3)) + 1):
current3 = 3**i3
if current3 > N2:
choice.append([3] * i3)
break
for i5 in range(ceil(log(N2, 5)) + 1):
current5 = current3 * 5**i5
if current5 > N2:
choice.append([3] * i3 + [5] * i5)
break
for i7 in range(ceil(log(N2, 7)) + 1):
current7 = current5 * 7**i7
if current7 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7)
break
for i9 in range(ceil(log(N2, 9)) + 1):
current9 = current7 * 9**i9
if current9 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7 +
[9] * i9)
break
for i11 in range(ceil(log(N2, 11)) + 1):
current11 = current9 * 11**i11
if current11 > N2:
choice.append([3] * i3 + [5] * i5 + [7] * i7 +
[9] * i9 + [11] * i11)
break
# print(len(choice))
# min_n = 10 ** 20 # inf
# min_comp = []
# for seq in choice:
# current_n = product(i**(j//2) for i, j in zip(primes, reversed(seq)))
# # print(current_n, seq)
# if current_n < min_n:
# min_n = current_n
# min_comp = seq
# print(min_comp)
min_n = min(
product(i**(j // 2) for i, j in zip(primes, reversed(seq)))
for seq in choice)
print(min_n)
def main():
ceiling = 5 * 10 ** 7
primes = prime_under(int((ceiling - 2**3 - 2**4) ** 0.5) + 1)
nums = set()
print("Primes generated by:", time() - starting_time, "seconds")
for i in primes:
for j in primes:
for k in primes:
num = i**4 + j**3 + k**2
if num > ceiling:
break
nums.add(num)
print(len(nums))
def main():
N = 10**5
# heuristic and slow!
print("\nheuristic and slow!")
D = 50 # probe depth
primes = prime_under(N)
ever = []
for p in primes[3:]:
if any(repunit_mod(p, i) == 0 for i in range(1, D + 1)):
# print(p)
ever.append(p)
continue
print(ever)
print(sum(primes) - sum(ever))
def main():
N = 10 ** 5
# heuristic and slow!
print("\nheuristic and slow!")
D = 50 # probe depth
primes = prime_under(N)
ever = []
for p in primes[3:]:
if any(repunit_mod(p, i) == 0 for i in range(1, D+1)):
# print(p)
ever.append(p)
continue
print(ever)
print(sum(primes) - sum(ever))
def main():
N = 4 * 10 ** 6
N2 = N * 2 + 1
primes = prime_under(50)
# print(log(N2, 3), len(primes), list(enumerate(primes)), sep='\n')
choice = []
for i3 in range(ceil(log(N2, 3))+1):
current3 = 3 ** i3
if current3 > N2:
choice.append([3]*i3)
break
for i5 in range(ceil(log(N2, 5))+1):
current5 = current3 * 5 ** i5
if current5 > N2:
choice.append([3]*i3 + [5]*i5)
break
for i7 in range(ceil(log(N2, 7))+1):
current7 = current5 * 7 ** i7
if current7 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7)
break
for i9 in range(ceil(log(N2, 9))+1):
current9 = current7 * 9 ** i9
if current9 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7 + [9]*i9)
break
for i11 in range(ceil(log(N2, 11))+1):
current11 = current9 * 11 ** i11
if current11 > N2:
choice.append([3]*i3 + [5]*i5 + [7]*i7 + [9]*i9 + [11]*i11)
break
# print(len(choice))
# min_n = 10 ** 20 # inf
# min_comp = []
# for seq in choice:
# current_n = product(i**(j//2) for i, j in zip(primes, reversed(seq)))
# # print(current_n, seq)
# if current_n < min_n:
# min_n = current_n
# min_comp = seq
# print(min_comp)
min_n = min(product(i**(j//2) for i, j in zip(primes, reversed(seq)))
for seq in choice)
print(min_n)
def main():
# print(m(2, 3, 1000))
N = 10000000
m_vals = []
primes = prime_under(N // 2 + 1)
primes_len = len(primes)
print("Primes computed by", time() - starting_time)
for i in range(primes_len):
if primes[i] * primes[i] > N:
break
for j in range(i + 1, primes_len):
if primes[i] * primes[j] > N:
break
m_curr = m(primes[i], primes[j], N)
# print(primes[i], primes[j], m_curr)
m_vals.append(m_curr)
print(sum(m_vals))
def main():
# print(m(2, 3, 1000))
N = 10000000
m_vals = []
primes = prime_under(N // 2 + 1)
primes_len = len(primes)
print("Primes computed by", time() - starting_time)
for i in range(primes_len):
if primes[i] * primes[i] > N:
break
for j in range(i + 1, primes_len):
if primes[i] * primes[j] > N:
break
m_curr = m(primes[i], primes[j], N)
# print(primes[i], primes[j], m_curr)
m_vals.append(m_curr)
print(sum(m_vals))
def main():
N = 10 ** 6
primes = prime_under(N + int(N ** 0.5))
print("Primes generated:", time() - starting_time)
sigma_s = 0
for i in range(3, len(primes)): # p1 starting from 5， p2 starting from 7
p1 = primes[i-1]
p2 = primes[i]
if p1 > N:
break
last_digits = 10 ** len(str(p1))
a = last_digits % p2
b = p2 - p1
curr_s = linear_mod(a, b, p2) * last_digits + p1
# print(p1, p2, curr_s)
sigma_s += curr_s
print(sigma_s)
def main():
N = 2 ** 30
sqrt_N = int(N ** 0.5) + 1
primes = prime_under(sqrt_N)
primes_len = len(primes)
print(primes_len, "primes generated by", time() - starting_time)
max_num_primes = 0
prime_product = primes[0]
while prime_product < sqrt_N:
max_num_primes += 1
prime_product *= primes[max_num_primes]
print("max number of primes", max_num_primes)
num_square_free = N
def square_free(start_index, level, last):
nonlocal num_square_free
for index in range(start_index, primes_len):
item = primes[index]
new_last = last * item
if new_last >= N:
break
num_square_free += N // (new_last * new_last) * (-1)**level
if level < max_num_primes:
square_free(index+1, level+1, new_last)
square_free(0, 1, 1)
# num_square_free = N
# flag = 1
# for num_primes in range(1, max_num_primes+1):
# print(num_primes, num_square_free, time() - starting_time)
# flag *= -1
# # print(generate_prime_tuple(primes, 0, num_primes, (), lambda l, a: product(l) * a < N))
# for prime_prod in generate_prime_prod_apply(primes, 0, num_primes, 1, lambda l, a: l*a < N, lambda a, b: a*b):
# squared = prime_prod ** 2
# num_square_free += N // squared * flag
print(num_square_free)
def main():
N = 2**30
sqrt_N = int(N**0.5) + 1
primes = prime_under(sqrt_N)
primes_len = len(primes)
print(primes_len, "primes generated by", time() - starting_time)
max_num_primes = 0
prime_product = primes[0]
while prime_product < sqrt_N:
max_num_primes += 1
prime_product *= primes[max_num_primes]
print("max number of primes", max_num_primes)
num_square_free = N
def square_free(start_index, level, last):
nonlocal num_square_free
for index in range(start_index, primes_len):
item = primes[index]
new_last = last * item
if new_last >= N:
break
num_square_free += N // (new_last * new_last) * (-1)**level
if level < max_num_primes:
square_free(index + 1, level + 1, new_last)
square_free(0, 1, 1)
# num_square_free = N
# flag = 1
# for num_primes in range(1, max_num_primes+1):
# print(num_primes, num_square_free, time() - starting_time)
# flag *= -1
# # print(generate_prime_tuple(primes, 0, num_primes, (), lambda l, a: product(l) * a < N))
# for prime_prod in generate_prime_prod_apply(primes, 0, num_primes, 1, lambda l, a: l*a < N, lambda a, b: a*b):
# squared = prime_prod ** 2
# num_square_free += N // squared * flag
print(num_square_free)
def main():
primes = prime_under(10 ** 7)
# print(len(primes), time() - starting_time)
factors = [2]
frontier = [4]
heapq.heapify(frontier)
prime_index = 1 # 2 already included
for i in range(500500 - 1):
if frontier[0] >= primes[prime_index]:
# push square of the current prime into the queue
heapq.heappush(frontier, primes[prime_index] ** 2)
factors.append(primes[prime_index])
prime_index += 1
else:
minimum = heapq.heappop(frontier)
heapq.heappush(frontier, minimum ** 2)
factors.append(minimum)
# print(factors)
print(product_mod(factors, 500500507))
def main():
N = 120000
primes = prime_under(N)
primes_len = len(primes)
pf_cache = [[0], [1]]
pf_cache.extend(prime_factors_under(N))
print("preprocessing finished", time() - starting_time, "seconds")
abc_hit = []
# 2: a = 1
for i in range(2, len(pf_cache)): # exclude 0, 1
if len(pf_cache[i]) == 1 and len(
pf_cache[i -
1]) == 1 and pf_cache[i][0] * pf_cache[i - 1][0] < i:
abc_hit.append(i)
# 3
print(len(list(rads3(N, primes, primes_len))))
for prod, ps in rads3(N, primes, primes_len):
pass
# 4
print(len(list(rads4(N, primes, primes_len))))
# 5
print(len(list(rads5(N, primes, primes_len))))
# 6
print(len(list(rads6(N, primes, primes_len))))
# for a, c in prod_cache:
# if prod_cache[a] * prod_cache[c-a] * prod_cache[c] < c:
# abc_hit.append(c)
# # print(a, c-a, c, pf_cache[c])
print(len(abc_hit))
print(sum(abc_hit))
print(abc_hit)
def main():
squbes200 = []
primes = prime_under(int(MAX ** 0.5))
prime_len = len(primes)
for i in range(prime_len):
if primes[i] ** 3 > MAX:
break
for j in range(i+1, prime_len):
sqube = primes[i] ** 3 * primes[j] ** 2
if sqube > MAX:
break
if '200' in str(sqube):
squbes200.append(sqube)
sqube = primes[i] ** 2 * primes[j] ** 3
if '200' in str(sqube) and sqube < MAX:
squbes200.append(sqube)
squbes200 = sorted(filter(lambda a: a < MAX, squbes200))
# print(squbes200[-1])
# print(len(squbes200))
print(list(islice((s for s in squbes200 if prime_proof(s)), 200))[-1])
def main():
squbes200 = []
primes = prime_under(int(MAX**0.5))
prime_len = len(primes)
for i in range(prime_len):
if primes[i]**3 > MAX:
break
for j in range(i + 1, prime_len):
sqube = primes[i]**3 * primes[j]**2
if sqube > MAX:
break
if '200' in str(sqube):
squbes200.append(sqube)
sqube = primes[i]**2 * primes[j]**3
if '200' in str(sqube) and sqube < MAX:
squbes200.append(sqube)
squbes200 = sorted(filter(lambda a: a < MAX, squbes200))
# print(squbes200[-1])
# print(len(squbes200))
print(list(islice((s for s in squbes200 if prime_proof(s)), 200))[-1])
def main():
N = 120000
primes = prime_under(N)
primes_len = len(primes)
pf_cache = [[0], [1]]
pf_cache.extend(prime_factors_under(N))
print("preprocessing finished", time() - starting_time, "seconds")
abc_hit = []
# 2: a = 1
for i in range(2, len(pf_cache)): # exclude 0, 1
if len(pf_cache[i]) == 1 and len(pf_cache[i-1]) == 1 and pf_cache[i][0] * pf_cache[i-1][0] < i:
abc_hit.append(i)
# 3
print(len(list(rads3(N, primes, primes_len))))
for prod, ps in rads3(N, primes, primes_len):
pass
# 4
print(len(list(rads4(N, primes, primes_len))))
# 5
print(len(list(rads5(N, primes, primes_len))))
# 6
print(len(list(rads6(N, primes, primes_len))))
# for a, c in prod_cache:
# if prod_cache[a] * prod_cache[c-a] * prod_cache[c] < c:
# abc_hit.append(c)
# # print(a, c-a, c, pf_cache[c])
print(len(abc_hit))
print(sum(abc_hit))
print(abc_hit)
"""
from mathutil import is_prime, prime_under
def is_truncatable(x):
s = str(x)
for i in range(1, len(s)):
if int(s[:i]) not in primes:
return False
for i in range(-1, -len(s), -1):
if int(s[i:]) not in primes:
return False
return True
primes = set(prime_under(800000))
def main():
truncatable = []
for prime in primes:
if is_truncatable(prime):
truncatable.append(prime)
if len(truncatable) == 11:
break
print(truncatable)
print(len(truncatable))
print(sum(truncatable))
from mathutil import is_prime, prime_under
def is_truncatable(x):
s = str(x)
for i in range(1, len(s)):
if int(s[:i]) not in primes:
return False
for i in range(-1, -len(s), -1):
if int(s[i:]) not in primes:
return False
return True
primes = set(prime_under(800000))
def main():
truncatable = []
for prime in primes:
if is_truncatable(prime):
truncatable.append(prime)
if len(truncatable) == 11:
break
print(truncatable)
print(len(truncatable))
print(sum(truncatable))