# Find number of circular primes below 1 million from pe_utils import primes_to_n, primality_test from math import log10 from functools import reduce primes = set(list(primes_to_n(1000000))) num_primes = len(primes) circular_primes = set() def all_rotations(n): yield n digits_m1 = int(log10(n)) ten_e_digits = 10**digits_m1 for i in range(digits_m1): n = n / 10 + n % 10 * ten_e_digits yield n def rotation_is_prime(rotations): for rotation in rotations: if rotation not in primes: return False return True for prime in primes: if prime not in circular_primes: rotations = list(all_rotations(prime)) if rotation_is_prime(rotations): for rotation in rotations: circular_primes.add(rotation)
# Find number of circular primes below 1 million
from pe_utils import primes_to_n, primality_test
from math import log10
from functools import reduce
primes = set(list(primes_to_n(1000000)))
num_primes = len(primes)
circular_primes = set()
def all_rotations(n):
yield n
digits_m1 = int(log10(n))
ten_e_digits = 10**digits_m1
for i in range(digits_m1):
n = n / 10 + n % 10 * ten_e_digits
yield n
def rotation_is_prime(rotations):
for rotation in rotations:
if rotation not in primes:
return False
return True
for prime in primes:
if prime not in circular_primes:
rotations = list(all_rotations(prime))
if rotation_is_prime(rotations):
# Find the prime generating polynomial of form n^2 + an + b
# where |a| < 1000 & |b| < 1000 and generates the longest array of primes
from pe_utils import prime_sieve, discriminant_bc, complex_quad_polys, primes_to_n
primes = list(prime_sieve(10000))
# only quadratics with disriminant less than 0 can be prime generating
# |b| <= 63
# c cannot be negative, otherwise discriminant would be positive
# 1 <= c <= 1000
# |c| must be prime (1st iteration is always c, and since we're finding primes, c must always be prime)
poly_gen = complex_quad_polys(range(-63, 63 + 1), list(primes_to_n(1000)))
def polynomial(a, b, i):
return i * i + a * i + b
highest_num_primes = 0
high_ab = None
for ab in poly_gen:
iterations = 0
while polynomial(ab[0], ab[1], iterations) in primes:
iterations += 1
if iterations > highest_num_primes:
high_ab = ab
highest_num_primes = iterations
print(high_ab, highest_num_primes)
# Find the prime generating polynomial of form n^2 + an + b # where |a| < 1000 & |b| < 1000 and generates the longest array of primes from pe_utils import prime_sieve, discriminant_bc, complex_quad_polys, primes_to_n primes = list(prime_sieve(10000)) # only quadratics with disriminant less than 0 can be prime generating # |b| <= 63 # c cannot be negative, otherwise discriminant would be positive # 1 <= c <= 1000 # |c| must be prime (1st iteration is always c, and since we're finding primes, c must always be prime) poly_gen = complex_quad_polys(range(-63, 63 + 1), list(primes_to_n(1000))) def polynomial(a, b, i): return i*i + a*i + b highest_num_primes = 0 high_ab = None for ab in poly_gen: iterations = 0 while polynomial(ab[0], ab[1], iterations) in primes: iterations += 1 if iterations > highest_num_primes: high_ab = ab highest_num_primes = iterations print(high_ab, highest_num_primes) print(highest_num_primes, high_ab)
