def prime_concat_partners(list_, primes, failure_point):
result = {}
length = len(list_)
for first in range(length - 1):
n1 = list_[first]
for second in range(first + 1, length):
n2 = list_[second]
cand1 = int(str(n1) + str(n2))
cand2 = int(str(n2) + str(n1))
if is_prime(cand1, primes=primes, failure_point=failure_point):
if is_prime(cand2, primes=primes, failure_point=failure_point):
# sets value to [] if not set, returns value at key
result.setdefault(n1, []).append(n2)
result.setdefault(n2, []).append(n1)
return result
def possible_pairings(partner_hash, length):
# length = 1
result = [[key] for key in partner_hash]
for size in range(2, length + 1):
next_iteration = []
for subset in result:
possible_additions = partner_hash[subset[0]]
for val in subset:
possible_additions = [entry for entry in possible_additions
if entry in partner_hash[val]]
next_iteration.extend([subset[:] + [candidate]
for candidate in possible_additions])
result = next_iteration
return result
def main(verbose=False):
MAX_n = 10**4
PRIMES = sieve(MAX_n)
partner_hash = prime_concat_partners(PRIMES, PRIMES, MAX_n**2)
valid = possible_pairings(partner_hash, 5)
min_sum = 10**10
min_set = None
for subset in valid:
if sum(subset) < min_sum:
min_sum = sum(subset)
min_set = subset
min_set = [str(prime) for prime in sorted(min_set)]
if verbose:
return '%s.\nThis is obtained with the primes %s.' % (
min_sum, ', '.join(min_set))
else:
return min_sum
if __name__ == '__main__':
print euler_timer(60)(main)(verbose=True)
def prime_concat_partners(list_, primes, failure_point):
result = {}
length = len(list_)
for first in range(length - 1):
n1 = list_[first]
for second in range(first + 1, length):
n2 = list_[second]
cand1 = int(str(n1) + str(n2))
cand2 = int(str(n2) + str(n1))
if is_prime(cand1, primes=primes, failure_point=failure_point):
if is_prime(cand2, primes=primes, failure_point=failure_point):
# sets value to [] if not set, returns value at key
result.setdefault(n1, []).append(n2)
result.setdefault(n2, []).append(n1)
return result
def main(verbose=False):
# layer/primes
# 2/3
# 1581/835
# 3536/1677
# 5000/2249
# 13121/5248, winning layer
# ratio >= .1 iff 10*(primes/total) >= 1 iff 10*primes >= total
# iff 10*primes >= 4*index - 3
FAILURE_POINT = 10 ** 9
PRIMES = sieve(int(sqrt(FAILURE_POINT)) + 1)
layer = 2
num_primes = 3
while 10 * num_primes >= 4 * layer - 3:
layer += 1
candidates = [(2 * layer - 1) ** 2 - 2 * (layer - 1) * i
for i in range(1, 4)]
if candidates[-1] >= FAILURE_POINT:
raise ValueError("Sieve was not big enough, restart function")
for candidate in candidates:
if is_prime(candidate, primes=PRIMES, failure_point=FAILURE_POINT):
num_primes += 1
side_length = 2 * layer - 1 # 2*(layer - 1) + 1
return side_length
def is_truncatable_prime(n, primes):
candidates = truncated_all(n)
for candidate in candidates:
if candidate in primes:
continue
elif is_prime(candidate):
primes.append(candidate)
else:
return False
return True
def main(verbose=False):
MAX_n = 987654321
PRIMES = sieve(int(sqrt(MAX_n)))
# A 9 digit pandigital will have digit sum 45, so can't be prime
# must be divisible by 9
for i in range(8, 1, -1):
cand = [str(dig) for dig in range(1, i + 1)]
cand = int("".join(cand))
candidates = sorted(all_permutations_digits(cand))[::-1]
for candidate in candidates:
if is_prime(candidate, primes=PRIMES, failure_point=MAX_n):
return candidate
raise ValueError("No prime was found, algorithm busted.")
