def truncatable_primes(limit=None):
"""Returns all truncatable primes up to limit, or optionally all 11."""
# This is similar to the circular primes problem in that we can eliminate
# a lot of primes that may have even numbers or 5s anywhere but the first
# digit.
primes = prime_sieve(limit) if limit else prime_sieve(1000000)
digits = '024568'
end_dg = '14689'
primes = [x for x in primes if not any([d in str(x)[1:] for d in digits])]
primes = [x for x in primes if not any([d in str(x)[0] for d in end_dg])]
primes = [x for x in primes if not any([d in str(x)[-1] for d in end_dg])]
primes = primes[4:]
p_gen = prime_generator(1000000)
trunc_primes = []
i = 0
while len(trunc_primes) < 11 and (i < len(primes) or not limit):
try:
p = str(primes[i])
except IndexError:
p = str(next(p_gen))
for j in range(1, len(p)):
if not is_prime(int(p[j:])) or not is_prime(int(p[:-j])):
break
else:
trunc_primes.append(int(p))
i += 1
return trunc_primes
def all_concat_prime_with(primes, new_element):
for prime in primes:
c = int(str(prime) + str(new_element))
if c not in cache:
cache[c] = is_prime(c, cached_primes)
if not cache[c]:
return False
c = int(str(new_element) + str(prime))
if c not in cache:
cache[c] = is_prime(c, cached_primes)
if not cache[c]:
return False
return True
def is_circular_prime(i):
circular_primes_list = [i]
for rotation in generate_rotations(i):
circular_primes_list.append(rotation)
if not is_prime(rotation):
return False
return circular_primes_list
def all_concat_prime(p):
for combination in itertools.permutations(p, 2):
c = int(str(combination[0]) + str(combination[1]))
if c not in cache:
cache[c] = is_prime(c)
if not cache[c]:
return False
return True
def is_truncatable_prime(prime):
if prime < 10:
return False
digits = str(prime)
while len(digits) > 1:
digits = digits[1:]
if not is_prime(int(digits)):
return False
digits = str(prime)
while len(digits) > 1:
digits = digits[:-1]
if not is_prime(int(digits)):
return False
return True
def find_n(a, b, max_n):
for n in xrange(0, max_n):
val = formula(a, b, n)
prime = is_prime(val)
if not prime:
return n - 1
return max_n
def prime_concats(n):
"""
Generates a dictionary of primes less than n with a list of other primes
that, when concatenated together, also form a prime.
"""
primes = prime_sieve(n)
prime_dict = {}
for i in range(len(primes)):
p = primes[i]
prime_dict[p] = []
for q in primes[i + 1:]:
if is_prime(int(str(p) + str(q))) and is_prime(
int(str(q) + str(p))):
prime_dict[p].append(q)
prime_concat_list = {}
for p in prime_dict:
if prime_dict[p] != []:
prime_concat_list[p] = prime_dict[p]
return prime_concat_list
def prime_generator(n):
"""Generates primes greater than n."""
while True:
if n % 6 == 1:
n += 4
elif n % 6 == 5:
n += 2
else:
n += 1
if is_prime(n):
yield n
def sum_primes(n):
"""
Sum up all prime numbers below n.
"""
sum = 0
for i in range(n):
# edge cases are handled by `is_prime'
if is_prime(i):
sum += i
return sum
def largest_pandigital_prime():
# The prime cannot have 9 digits, because the sum of the numbers 1 through
# 9 is divisible by 9, and hence will never be prime. The same logic holds
# for 8, as eliminating 9 from a number divisible by 9 is still divisible.
pandigitals = pandigital_generator(7, True)
while True:
p = next(pandigitals)
if str(p)[-1] in '024568':
continue
if is_prime(p):
break
return p
def prime_sum(limit):
if limit < 2: raise ValueError("limit must be >= 2")
primeSum = 2
primes = [2]
i = 3
while True:
if is_prime(i, primes):
primeSum += i
primes.append(i)
i += 2
if i > limit:
return primeSum
def diagonals(n):
if n <= 0:
return ([1], [])
if n in cache:
return cache[n]
o, p = diagonals(n - 1)
o = o[:] # copy cached list
p = p[:] # copy cached list
for i in range(4):
u = o[-1] + n * 2
if is_prime(u):
p.append(u)
o.append(u)
result = (o, p)
cache[n] = result
return result
def longest_quadratic_primes(limit):
b_list = prime_sieve(limit) # Because f(0) = b, so b must be prime.
# Further, when n = 1, p + 1 + even number will produce a non-prime. So a
# must be odd. The only exception would be if b = 2, in which case if n = 2,
# n² = 4 and a must again be odd to produce any more primes.
a_list = [x for x in list(range(-limit + 1, limit)) if x % 2 == 1]
max_length, max_a, max_b = 0, 0, 0
for a in a_list:
for b in b_list:
n = 0
while is_prime(n**2 + a * n + b):
n += 1
if n > max_length:
max_length = n
max_a = a
max_b = b
return (max_a, max_b)
def nth_prime(n):
"""
Determine the n'th prime number using a brute-force
approach.
"""
if n == 1:
return 2
current_num = 3
num_primes = 1 # 2 is already included
while True:
if is_prime(current_num):
num_primes += 1
if num_primes == n:
return current_num
current_num += 2 # even numbers aren't prime!
def solve():
i = 7.0
while True:
i += 2.0
if is_prime(i):
continue
for prime in generate_primes():
if prime > i:
return i
r = i - prime
if not r % 2 == 0:
continue
r = math.sqrt(r / 2.0)
if r == math.floor(r):
break
pass
def solve(limit=1000, prec=50, debug=False):
longest = 0
longest_i = 0
primes = [2]
for i in xrange(2, limit + 1):
if debug and i % 10 == 0:
print ".",
sys.stdout.flush()
if is_prime(i, primes):
num = precise_quotient(1, i, prec)
length = longest_recurring_cycle(num.replace("0.", ""))
if length > longest:
longest = length
longest_i = i
if debug:
print("")
print("longest: 1/" + str(longest_i) + " with len = " + str(longest))
return longest
def test_is_prime(self):
self.assertFalse(problem003.is_prime(1))
self.assertTrue(problem003.is_prime(2))
self.assertFalse(problem003.is_prime(4))
self.assertTrue(problem003.is_prime(7))
self.assertFalse(problem003.is_prime(20))
