def diagonal_sum():
primes = 0
not_primes = 0
for i in range(1, 28000):
top_right = ((2 * i)**2) - (2 * i) + 1
top_left = ((2 * i)**2) + 1
bottom_right = ((2 * i) - 1)**2
bottom_left = (((2 * i) + 1)**2) - ((2 * i) + 1) + 1
if is_prime(top_right):
primes += 1
else:
not_primes += 1
if is_prime(top_left):
primes += 1
else:
not_primes += 1
if is_prime(bottom_right):
primes += 1
else:
not_primes += 1
if is_prime(bottom_left):
primes += 1
else:
not_primes += 1
if primes / (not_primes + primes) <= 0.1:
return 2 * i + 1
def is_family(f):
for i in range(len(f)):
for j in range(i + 1, len(f)):
left = int_concat(f[i], f[j])
right = int_concat(f[j], f[i])
if not (is_prime(left) and is_prime(right)):
return False
return True
def is_truncatable_prime(number):
if not is_prime(number):
return False
number = str(number)
for x in range(1, len(number)):
if not is_prime(int(number[:-x])) or not is_prime(int(number[x:])):
return False
return True
def test_goldbach(n):
if n % 2 == 0 or helpers.is_prime(n):
return True
for i in itertools.count(1):
k = n - 2 * i * i
if k <= 0:
return False
elif helpers.is_prime(k):
return True
def good_pair(a, b):
key_pair = (a, b)
if key_pair in pairmem:
return pairmem[key_pair]
stra = str(a)
strb = str(b)
result = is_prime(int(stra + strb)) \
and is_prime(int(strb + stra))
pairmem[key_pair] = result
return result
def get_concat_primes(i): prime = primes[i] prime_concats = set() for j in xrange(i+1, len(primes)): prime_j = primes[j] c1 = concat(prime, prime_j) c2 = concat(prime_j, prime) c1_prime = c1 in primes if c1 < limit else is_prime(c1) c2_prime = c2 in primes if c2 < limit else is_prime(c2) if c1_prime and c2_prime: prime_concats.add(j) return prime_concats
def is_truncatable_prime(n):
# Left truncatable
i = 10
while i <= n:
if not helpers.is_prime(n % i):
return False
i *= 10
# Right truncatable
while n > 0:
if not helpers.is_prime(n):
return False
n //= 10
return True
def get_more_primes():
global working_combos, all_primes
all_primes = get_prime_numbers_until(10000)
all_primes.remove(2)
all_primes.remove(5)
for cur_prime in all_primes:
for prime in list(working_combos.keys()):
left = int(str(prime) + str(cur_prime))
right = int(str(cur_prime) + str(prime))
if is_prime(left) and is_prime(right):
working_combos[prime].add(cur_prime)
working_combos[cur_prime].add(prime)
working_combos[cur_prime] = set()
def truncate_left(n):
if not is_prime(n):
return False
else:
i = str(n)
while len(i) != 1:
i = int(i[1:])
if not is_prime(i):
return False
else:
i = str(i)
if len(i) == 1:
return True
def yield_circ_primes(limit):
for i in range(3, limit):
if is_prime(i):
m = list(str(i))
if rotate(m, i):
yield i
def result(n):
_max = 1
_max_a = 1
_max_b = 1
first_primes = [j for j in range(n + 1) if is_prime(j)]
for a in range(- n, n + 1):
for b in first_primes:
i = 0
while is_prime(binome(a, b)(i)):
i += 1
if i >= _max:
_max = i
_max_a = a
_max_b = b
return _max_a * _max_b
def p50():
primelist = list(primes(4000))
for x in range(1,200):
for y in range(x,-1,-1):
s = sum(primelist[y:len(primelist)-(x-y)])
if is_prime(s):
return s
def eight_prime_family(prime, rd):
c = 0
for digit in '0123456789':
n = int(str.replace(prime, rd, digit))
if n > 100000 and is_prime(n):
c += 1
return c == 8
def largest_prime_factor(n=600851475143):
"""P3. Finds the largest prime factor of n."""
for i in xrange(1, n, 2):
if n % i == 0:
# use its larger factor pair
j = n // i
if is_prime(j):
return j
def solution1(n):
"""
Slow solution. Performs primarly test in a range and checks
if it is a factor of n, divides n by it, and repeats until
factor of n is 1.
"""
current = n
max_factor = 1
while not is_prime(current) and current > 1:
for number in range(int(sqrt(current))):
if is_prime(number) and current % number == 0:
current = current / number
if number > max_factor:
max_factor = number
return max_factor
def p51():
primelist = [p for p in primes(1000000) if p > 100001]
for prime in primelist:
for number_family in number_familes(prime):
prime_family = [n for n in number_family if is_prime(n) and len(str(n)) == len(str(prime))]
if len(prime_family) == 8 and prime in prime_family:
print prime
return
def prime_factors(num):
"""Get prime factors of a number
My first attempt
"""
prime_factors = []
for i in range(2, num + 1):
if (num % i) == 0 and is_prime(i) == True:
prime_factors.append(i)
return prime_factors
def problem7():
""" The 10001st prime number"""
primelist = [2, 3]
i = 5
while len(primelist) < 10001:
if h.is_prime(i):
primelist.append(i)
i = i + 2 # increment by 2, to check odds only
return primelist[-1]
def answer(limit):
num_primes = 0
corner = 1
for i in xrange(1, limit):
side_length = 2 * i + 1
for _ in xrange(4):
corner = corner + side_length - 1
num_primes += 1 if is_prime(corner) else 0
if num_primes / (4.0 * i + 1.0) < 0.1:
return side_length
def main():
_len = 1
nb_primes = 0
ratio = 1
nb_diag_number = 1
while ratio > 0.1 or _len < 9:
_len += 2
_len2 = _len ** 2
if is_prime(_len2):
nb_primes += 1
if is_prime(_len2 - (_len - 1)):
nb_primes += 1
if is_prime(_len2 - 2 * (_len - 1)):
nb_primes += 1
if is_prime(_len2 - 3 * (_len - 1)):
nb_primes += 1
nb_diag_number += 4
ratio = nb_primes / nb_diag_number
return _len
def compute(num_digits):
# Option to lop off the start when using smaller pandigital numbers.
num = '987654321'[9 - num_digits:]
return list(
islice(
("".join(digit)
for digit in permutations(num) if is_prime(int("".join(digit)))),
1))
def nth_prime(n):
num = 1
prime = 1
prime_count = 0
while prime_count <= n:
if is_prime(num):
prime = num
prime_count += 1
num += 2
return prime
def try_prime(number):
truncatable_primes = []
if number > 9 and is_truncatable_prime(number):
truncatable_primes.append(number)
new_value = number * 10
for digit in digits:
new_prime = new_value + digit
if is_prime(new_prime):
truncatable_primes += try_prime(new_value + digit)
return truncatable_primes
def compute(limit):
n_max = 0
for b in prime_sieve(limit):
for a in range(-b, 0, 2):
n = 1
while m(n, a, b) > 0 and is_prime(m(n, a, b)):
n += 1
if n > n_max:
n_max, prod = n, a * b
return prod
def nth_prime1(n):
"""
Loop through numbers and use trial division for primarly test,
keep count of which ones are prime, return the nth prime.
"""
count = 0
current = 1
while count < n:
current += 1
if is_prime(current):
count += 1
return current
def rotate(m, n):
c = 0
while n != c:
end = m[1:]
end.append(m[0])
whole = ''.join(end)
c = int(whole)
if not is_prime(c):
return False
m = list(str(whole))
return True
def compute():
# Note: The only 1-digit pandigital number is 1, which is not prime. Thus we require n >= 2.
for n in reversed(range(2, 10)):
arr = list(reversed(range(1, n + 1)))
while True:
if arr[-1] not in NONPRIME_LAST_DIGITS:
n = int("".join(str(x) for x in arr))
if helpers.is_prime(n):
return str(n)
if not prev_permutation(arr):
break
raise AssertionError()
def answer(n):
# could speed up by iterating backwards in a permutation
trunc = pan[:n]
permutations = itertools.permutations(trunc)
max_val = 0
for p in permutations:
p_str = "".join(p)
val = int(p_str)
if is_prime(val) and is_pandigital(p_str):
max_val = max(max_val, val)
if max_val != 0:
return max_val
return answer(n - 1)
def problem3(N=600851475143):
"""What is the largest prime factor of the number N = 600851475143 ?
This solution will make use of the fact that a number is prime if it
has no prime factors smaller than sqrt(n). Potential factors are checked
from high to low, decrementing by 2 because N is not even.
"""
check_factor = int(sqrt(N))
while True:
if h.divisible(N,check_factor) and h.is_prime(check_factor):
return check_factor
else:
check_factor -= 1
def get_prime_factors(number):
"""Return all prime factors which must be multiplied
to generate a certain number
"""
factors_list = []
for j in range(2, number + 1):
while True:
if is_factor(number, j) and is_prime(j):
factors_list.append(j)
number = number // j
else:
break
return factors_list
def truncate_right(n):
j = str(n)
while len(j) != 1:
j = int(j[:-1])
if not is_prime(j):
return False
else:
j = str(j)
if len(j) == 1:
return True
else:
return False
def problem03():
"""
Q: What is the largest prime factor of the number 600851475143?
A: 6857
"""
n = 600851475143
factors = helpers.get_factors(n)
largest = 0
for factor in factors:
if factor != n and factor > largest and helpers.is_prime(factor):
largest = factor
return largest
def main():
# combinations of prime under 100 that are prime. 2, 3, 5 and 7 are skipped
truncatable_primes = [23, 37, 53, 73]
n = 101
offset = 1
while len(truncatable_primes) < 11:
# alternative to use accumulate([2, 1, 2], cycle([2, 4]))
n += 3 - offset # 2 or 4
offset *= -1
# all truncatables number above 100 cannot contains even digits or 5
if not re.search('[245680]', str(n)) and is_prime(n) and is_truncatable_prime(n):
truncatable_primes.append(n)
return sum(truncatable_primes)
def phi2(num):
# print [i for i in range(2,num) if areRelativePrime(num , i)]
if helpers.is_prime(num):
return -1
global maxPhi
phi_num = 1
phi_cache = []
for i in range(2, num):
for j in phi_cache:
if i % j == 0:
break
else:
if helpers.are_relative_prime(num, i):
phi_num += 1
if (num / phi_num) <= maxPhi:
return -1
else:
phi_cache.append(i)
return phi_num
def find_prime_ratio(required_ratio, upper_bound):
s = 1
prime_count = 0
primes = primes_up_to(upper_bound)
for diagonals in get_diagonals():
for d in diagonals:
if d > upper_bound:
if is_prime(d, cache_result=False):
prime_count += 1
elif d in primes:
prime_count += 1
ratio = (prime_count / float(s * 4))
if ratio < required_ratio:
return (2 * s) + 1
s += 1
def main():
list_primes = list(eratosthene(1000000))
for prime in list_primes:
if prime < 100000:
continue
dict_count = get_digit_count_dict(prime)
for _str in ['0', '1', '2']:
if dict_count[_str] >= 3:
size = 0
error = 0
# replace all numbers
mask = str(prime).replace(_str, '.')
for other_digit in range(int(_str), 10):
if error > 3:
break
if is_prime(int(mask.replace('.', str(other_digit)))):
size += 1
else:
error += 1
if size == 8:
return prime
def compute(limit):
for k in range(limit):
if is_prime(k) and truncate_left(k) and truncate_right(k):
yield k
import time, math, helpers
start = time.clock()
primecount = 0
totcount = 1
num = 1
for i in range(2, 100000, 2):
for j in range(4):
if helpers.is_prime(num):
primecount += 1
totcount += 1
num += i
if float(primecount)/float(totcount)<0.10:
print i+1
print time.clock()-start
quit()
def is_harshad(num):
if num < 10 or num in hasdasdfs:
return True
if num in not_hasss:
return False
if num % sum([int(i) for i in str(num)]) == 0:
return is_harshad(num // 10)
return False
num = 1
hazards = {0: [1, 2, 3, 4, 5, 6, 7, 8, 9]}
results = []
for k in range(0, 12):
hazards[k + 1] = []
for h in hazards[k]:
h *= 10
for i in range(0, 10):
hi = h + i
if is_harshad(hi):
hazards[k + 1].append(hi)
if helpers.is_prime(hi / sum([int(ii) for ii in str(hi)])):
for p in range(0, 10):
the_num = hi * 10 + p
if helpers.is_prime(the_num):
results.append(the_num)
# print("\033[0;35mi", hi, "\033[0m")
print("\033[0;35msum(results)", sum(results), "\033[0m")
print(time.clock() - start_time)
def concat(i,j):
if not is_prime(int(str(i)+str(j))) or not is_prime(int(str(j)+str(i))):
return False
return True
def is_truncatable_prime(n):
for d in range(1, len(str(n))):
if not is_prime(int(str(n)[d:])) or not is_prime(int(str(n)[:d])):
return False
return True
def solution():
total = 0
for num in range(LIMIT):
if is_prime(num):
total += num
return total
'''
Created on Jan 9, 2017
@author: jfinn
'''
from helpers import is_prime
from _functools import reduce
import operator
primes = [2]
for n in range(3, 2000000):
if is_prime(n, primes):
primes.append(n)
print("Found", len(primes), "primes below 2,000,000")
print(reduce(operator.add, primes))
#!env/bin/python
import helpers as h
c = 0
x = 23
r = 0
while c < 11:
while not h.is_prime(x):
x += 2
for ch in str(x):
if int(ch) % 2 == 0 and ch != '2':
x += 2
number = x
skip = False
while number > 0:
if not h.is_prime(number):
skip = True
break
number /= 10
if skip:
x += 2
continue
number = x % (10**int(h.log10(x)))
while number > 0:
if not h.is_prime(number):
skip = True
break
number = number % (10**int(h.log10(number)))
if skip:
#!env/bin/python
import helpers as h
MAX = 1000
bl = -1
ba = 0
bb = 0
br = -1
for a in range(-1*MAX,MAX+1):
for b in range(-1*MAX+1,MAX+1,2):
if not h.is_prime(b):
continue
i = 0
r = (i*i)+(a*i)+b
pr = 0
while h.is_prime(r):
i += 1
r = (i*i)+(a*i)+b
if i >= bl:
bl = i
ba = a
bb = b
print 'n = 0 to '+str(bl)
print 'a = '+str(ba)
print 'b = '+str(bb)
def a_n(n): if n == 1: return 0 elif is_prime(n): return 1 else: return 0
#!env/bin/python
import helpers as h
def list_to_num(n):
s = ''
for x in n:
s = s + str(x)
return int(s)
result = None
s = [10,9,8,7,6,5,4,3,2,1]
while result == None:
s.pop(0)
t = 0
for x in s:
t += x
if t % 3 == 0:
continue
for x in h.permute(s):
if x[len(s)-1] % 2 == 0:
continue
if h.is_prime(list_to_num(x)):
result = x
break
print s
print result
def concat(lijst):
for i, j in combinations(lijst, 2):
if not is_prime(int(str(i) + str(j))) or not is_prime(
int(str(j) + str(i))):
return False
return True
from helpers import sieve, is_prime
p = sieve(10000)
total = 0
start = 0
length = 0
maxlen = 0
while True:
if total + p[start + length] < 1000000:
total += p[start + length]
length += 1
if length > maxlen and is_prime(total):
if 1000000 - total < p[start + length]:
print total
break
maxlen = length
else:
total -= (p[start] + p[start + length - 1])
length -= 2
start += 1
def get_prime_factors(num):
"""Returns the prime factors for num."""
for i in xrange(2, num):
if num % i == 0 and is_prime(i):
yield i
def prime_factors(number):
"""A better version utilizing reduce to compute
the factors
"""
all_factors = factors(number)
return list(filter(lambda x: is_prime(x), all_factors))
from helpers import is_prime
size = 1
value = 1
total = 1
primes = 0
while True:
size += 2
top_right_value = value + size - 1
top_left_value = top_right_value + size - 1
bottom_left_value = top_left_value + size - 1
bottom_right_value = bottom_left_value + size - 1
value = bottom_right_value
if is_prime(top_right_value):
primes += 1
if is_prime(top_left_value):
primes += 1
if is_prime(bottom_left_value):
primes += 1
if is_prime(bottom_right_value):
primes += 1
total += 4
if primes / total < 0.1:
break
print(size)
# This is really dumb and takes forever to run, I know # A smarter way would just check random combos matching ([3,7,9][1,3,7,9]*[3,7,9]) until you find 11 from helpers import sieve, is_prime print sum([int(p) for p in map(str,sieve(1000000)[4:]) if all([is_prime(int(p[i:])) and is_prime(int(p[:i+1])) for i in range(len(p))])])
def main(): filtered = [i for i in range(3, NUMBER, 2)] print (2 + sum(i for i in filtered if is_prime(i)))
def jamcoin_has_prime_base(self, jamcoin_bases):
print jamcoin_bases
return [base for base in jamcoin_bases if helpers.is_prime(base)]
'''
Created on Jan 22, 2017
@author: jfinn
'''
from helpers import is_prime
primes = [2]
for num in range(3, 1000000):
if is_prime(num):
primes.append(num)
primes = set(primes) # faster lookup
circ_primes = []
num_circ = 0
for prime in primes:
is_circ_prime = True
num_digits = len(str(prime))
num = prime
for i in range(num_digits):
if num not in primes:
is_circ_prime = False
break
ones = num % 10
num = (num // 10) + (ones * 10**(num_digits - 1))
if is_circ_prime:
num_circ += 1
circ_primes.append(prime)
print(circ_primes)