def first_approach():
coefficients = []
for a in range(-999, 1000):
for b in range(-1000, 1001):
if not is_prime(euler_polynomial(a, b, 0)):
continue
else:
x = 1
while is_prime(euler_polynomial(a, b, x)):
x += 1
coefficients.append((a, b, x))
a, b, _ = max(coefficients, key=lambda k: k[2])
print('Solution =', a * b)
def generate_candidates_from_prime(prime):
candidates = set()
for candidate in permutations(str(prime)):
candidate = int(''.join(candidate))
if is_prime(candidate) and log10(candidate) >= 3:
candidates.add(candidate)
return candidates
def sum_strong_right_truncatable_harshad_primes():
result = 0
for h in takewhile(lambda x: x < LIMIT,
right_truncatable_harshad_numbers()):
if is_strong(h):
for d in range(1, 10, 2):
candidate = 10 * h + d
if is_prime(candidate):
result += candidate
return result
def attempt():
result = 0
for places in range(2, 10):
print(places, 'places')
for pandigital in permutations('123456789'[:places]):
pandigital = int(''.join(pandigital))
if is_prime(pandigital):
print(' ', pandigital)
result = max(result, pandigital)
return result
def attempt():
limit = 10**6
prime = 0
terms = 1
prime_sums = [0]
for p in prime_number_generator():
prime_sums.append(prime_sums[-1] + p)
if prime_sums[-1] >= limit:
break
candidates = []
for j in range(len(prime_sums)):
for k in range(j):
candidate = prime_sums[j] - prime_sums[k]
if is_prime(candidate):
candidates.append((candidate, j - k))
break
return max(candidates, key=lambda x: x[terms])[prime]
def is_strong(h):
return is_prime(h / digit_sum(h))
def count_primes(set_of_numbers):
return len({p for p in set_of_numbers if is_prime(p)})
"""Detect primitive roots of a prime module."""
import sys
from mathext import euler_phi, relative_primes, is_prime
MODULE = 99989
# MODULE = 31
if __name__ == '__main__':
if not is_prime(MODULE):
sys.exit()
print('Module:', MODULE)
print('This module has', euler_phi(MODULE - 1), 'primitive roots.')
print('Probing the first primitive root ...')
for a in range(2, MODULE - 1):
print('Caandidate:', a)
for k in range(1, MODULE - 1):
x = pow(a, k, MODULE)
if k < (MODULE - 1) and x == 1:
print(a, 'is not a primitive root.')
break
else:
print(a, 'is a primitive root for module', MODULE)
pw0 = a
break
rp = relative_primes(MODULE - 1)
print('The set of totients of the totient count is:', rp)
pw = []
for k in rp:
pw.append(pow(pw0, k, MODULE))
print('Primitive roots:', sorted(pw))