def IsPrimeTruncatable(n, ndigits):
if (n > 100):
if (prime.isprime(n) == False):
for i in range(2, ndigits):
if (prime.isprime(n // 10**i) == False
or prime.isprime(n % 10**i) == False):
return False
def N(d,n=10):
"""For an n-digit prime, d is the digit that reappears."""
primes,dict = [], M()
if dict[d] == n-1:
to_permute = list(repeat(str(d),n)) #['1','1','1','1']
digits = [str(k) for k in range(10) if k!=d]
current_permute = int("".join(to_permute))
if prime.isprime(current_permute): primes.append(current_permute)
for i in range(n):
for digit in digits:
to_permute[i] = digit
cur_num = int("".join(to_permute))
if prime.isprime(cur_num):
primes.append(cur_num)
to_permute[i] = str(d)
return primes
if dict[d] == n-2:
to_permute = list(repeat(str(d),n)) #['1','1','1','1']
digits = [str(k) for k in range(10) if k!=d]
double_digits = list(product(digits,digits))
current_permute = int("".join(to_permute))
if prime.isprime(current_permute): primes.append(current_permute)
for i in range(n-1):
for j in range(1,n):
for pair in double_digits:
to_permute[i] = pair[0]
to_permute[j] = pair[1]
cur_num = int("".join(to_permute))
# print "on",cur_num
if prime.isprime(cur_num):
# print cur_num,"PRIME"
primes.append(cur_num)
to_permute[i] = str(d)
to_permute[j] = str(d)
return [k for k in primes if len(str(k))==n]
def l_r_primes(n):
prime_left = prime_right = False
for i in xrange(1, len(str(n))):
l = n % pow(10, i)
r = n / pow(10, i)
if (not prime.isprime(l)) or (not prime.isprime(r)):
return False
return True
def test_time_taken_to_calculate():
t0 = time.time()
for i in range(2, 100000):
prime.isprime(i)
t1 = time.time()
t = t1 - t0
assert t < 0.4
def answer():
# observation: cannot be 8 or 9 due to divisibility by 3
for n in range(7, 0, -1):
possibilities = map(tuple_to_int, permutations(range(n, 0, -1)))
for candidate in possibilities:
if isprime(candidate):
return candidate
def f():
for i in reversed(range(1, 547)):
for j in range(1, 549-i):
print j, i
if isprime(sum(islice(sieve, j, i))):
print j-i+1
return sum(islice(sieve, j, i))
def bruteforce():
count=1 # 1/2 is special
for d in range(3,N+1):
if prime.isprime(d):
count+=d-1
else:
count+=count_bruteforce(d) # error
def answer(limit=1000000):
_refresh(limit // pow(10, sqrt(log10(limit))-1))
for length in range(int(sqrt(limit)), 0, -1):
for end in range(len(prime_list), length-1, -1):
candidate = sum(prime_list[end-length:end])
if candidate < limit and isprime(candidate):
return candidate
def answer(limit=1000000):
_refresh(limit // pow(10, sqrt(log10(limit)) - 1))
for length in range(int(sqrt(limit)), 0, -1):
for end in range(len(prime_list), length - 1, -1):
candidate = sum(prime_list[end - length:end])
if candidate < limit and isprime(candidate):
return candidate
def check_length(n, below):
maxprime = 0
for x in xrange(0, below):
total = sum(prime.prime_list[x:x+n])
if total > below: break
if prime.isprime(total): maxprime = total
return maxprime
def gen_pnr(size, m):
while True:
# 1 шаг
p = prime.gen_prime(size)
assert p % 4 == 1
assert sympy.isprime(p)
# 2 шаг (страница 168, алгоритм 7.8.1)
a, b = get_zi_factors(p, 1)
assert a * a + b * b == p
# 3 шаг
t_set = [2 * a, -2 * a, 2 * b, -2 * b]
for t in t_set:
n = p + 1 + t
rs = [n // 2, n // 4]
for r in rs:
if prime.isprime(r):
assert sympy.isprime(r)
# шаг 4
if p != r:
for i in range(1, m + 1):
if pow(p, i, r) == 1:
break
else:
return p, n, r
def answer():
# observation: cannot be 8 or 9 due to divisibility by 3
for n in range(7, 0, -1):
possibilities = map(tuple_to_int, permutations(range(n, 0, -1)))
for candidate in possibilities:
if isprime(candidate):
return candidate
def ProjectEuler37():
"""
The number 3797 has an interesting property. Being prime itself,
it is possible to continuously remove digits from left to right,
and remain prime at each stage: 3797, 797, 97, and 7.
Similarly we can work from right to left: 3797, 379, 37, and 3.
Find the sum of the only eleven primes that are both truncatable from left to right and right to left.
NOTE: 2, 3, 5, and 7 are not considered to be truncatable primes.
Solution:
1 is not prime, so two digits can only be 37 73
then, for number has 3 or more digits, those must be started and ended with 3 or 7
but the digits between can be combination_with_repeation of [1,3,7,9]
"""
count = 2
numbers = [
37, 73, 23, 53
] # I miss the first two 23, 35, filter the 2digits combination of [2 3 5 7]
MaxDigits = 10
TotalNo = 11
for d in range(3, MaxDigits + 1):
# generate a list of possible numbers
for n in ListOfCandiates(d):
if (all(prime.isprime(i) for i in ListOfTruncatble(n, d))):
count = count + 1
numbers.append(n)
print(n)
if (len(numbers) == TotalNo): break
if (len(numbers) == TotalNo): break
print(numbers)
print(sum(numbers))
def problem10():
"""
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
Answer: 142913828922
"""
from prime import isprime
print("problem10:", sum([i for i in range(2, 2000000) if isprime(i)]))
def is_left_truncatable(i):
while True:
if prime.isprime(i):
if i > 10:
i = int("".join(list(str(i))[1::]))
else:
return True
else:
return False
def countProperFaction(d):
""" for non-prime number, get the list of factor
proper factions for denumerator no bigger than 1000: 304191 (tested correctly)
"""
if prime.isprime(d):
return (d-1)
else:
#return count_bruteforce(d)
return factorset_union(d)
def problem58():
numOnDiagonal, width, noOfPrimes = 13, 7, 8
while noOfPrimes*10 > numOnDiagonal:
nextWidth, square = width+2, width*width
nextSquare = nextWidth*nextWidth
result = xrange(square+width+1, nextSquare+1, width+1)
noOfPrimes = noOfPrimes+ len(filter(lambda n: prime.isprime(n), result))
numOnDiagonal = numOnDiagonal+4
width = nextWidth
return width
def answer(alim=1000, blim=1000):
maxn = 0
for a in range(-alim+1, alim):
for b in range(-blim+1, blim):
n = 0
while isprime(pow(n, 2) + a*n + b):
n += 1
if n > maxn:
maxa, maxb, maxn = a, b, n
return maxa * maxb
def main():
i = 11
t = 10
res = 0
while i > 0:
if prime.isprime(t) and l_r_primes(t):
res += t
i -= 1
t += 1
print res
def answer():
for candidate in count(3, 2):
if isprime(candidate):
continue
for p in prime_list:
remainder = candidate - p
root = sqrt(remainder // 2)
if root == int(root):
break
else:
return candidate
def answer():
for candidate in count(3, 2):
if isprime(candidate):
continue
for p in prime_list:
remainder = candidate - p
root = sqrt(remainder//2)
if root == int(root):
break
else:
return candidate
def phi(n):
""" if it is prime number, n/phi(n) is small near 1
but for prime number phi(n)=n-1,
"""
if isprime(n):
return n - 1
else:
s = []
#to-do
return len(s)
def is_right_truncatable(i):
while True:
if prime.isprime(i):
if i > 10:
a = list(str(i))
a.pop()
i = int("".join(a))
else:
return True
else:
return False
def f(n):
result = []
number = 0
for i in count(91):
if i%2 == 0 or i%5 == 0:
continue
if (not isprime(i)) and is_ok(i):
result.append(i)
number += 1
if number == n:
break
return sum(result)
def pandigital(order):
nums = [str(i) for i in xrange(1, order+1)]
product = []
for num in per(nums):
num = ''.join(num)
if isprime(int(num)) and check(num, order):
product.append(int(num))
else:
continue
if product:
return max(product)
else:
return None
def prime_family_length(n, digits):
if cache.has_key((n, digits)): return cache[n, digits]
num, nums, count = list(str(n)), [], 0
if len(dict.fromkeys(num[d] for d in digits).keys()) > 1:
return cache.setdefault((n, digits), 0) # The digits must have the same number
for d in range(0 in digits and 1 or 0, 10): # Ensure 0 is not the first digit
for x in digits: num[x] = str(d)
n = int(''.join(num))
if prime.isprime(n): count += 1
nums.append(n)
for n in nums: cache[n, digits] = count
return count
def e58():
maxnum = 20000
prime_count = 0
count = 1
for n in xrange(2, maxnum):
count += 4
x = (2 * n - 1) ** 2
for m in range(1, 4):
if prime.isprime(x - 2 * (n - 1) * m):
prime_count += 1
ratio = float(prime_count) / count
if ratio <= 0.1:
print 2 * n - 1, prime_count, '/', count, '=', ratio
break
def answer():
for numbers in combinations_with_replacement(range(10), 4):
perms_l = [digits_to_num(n) for n in permutations(numbers)]
perms_s = set(perms_l)
for fst_index, fst in enumerate(perms_l):
if fst <= 1487: # rule out given value
continue
for snd_index in range(fst_index + 1, len(perms_l)):
snd = perms_l[snd_index]
dif = snd - fst
if dif == 0:
continue
thrd = snd + dif
if thrd in perms_s:
if all(isprime(n) for n in (fst, snd, thrd)):
return fst, snd, thrd
def answer():
for numbers in combinations_with_replacement(range(10), 4):
perms_l = [digits_to_num(n) for n in permutations(numbers)]
perms_s = set(perms_l)
for fst_index, fst in enumerate(perms_l):
if fst <= 1487: # rule out given value
continue
for snd_index in range(fst_index+1, len(perms_l)):
snd = perms_l[snd_index]
dif = snd-fst
if dif == 0:
continue
thrd = snd+dif
if thrd in perms_s:
if all(isprime(n) for n in (fst, snd, thrd)):
return fst, snd, thrd
#!/usr/bin/python3
import prime
MAX = 10000
prime._refresh(MAX)
squares = dict.fromkeys((x*x for x in range(1, MAX)), 1)
for x in range(35, MAX, 2):
if not prime.isprime(x):
is_goldbach = 0
for p in prime.prime_list:
if p >= x: break
key = (x-p)/2
if key in squares:
is_goldbach = 1
break
if not is_goldbach:
print(x)
break
from math import sqrt
import prime
primes = filter(lambda k: prime.isprime(k), xrange(2,51))
mult = lambda x: reduce(lambda y, z: y * z, x, 1)
translate = lambda x: mult(primes[i] ** ((x[i] - 1) // 2) for i in range(len(x)))
def generator(limit):
l = [1] * 14
while l[13] < limit:
i = 13
while i > 0 and (l[i] == l[i-1]):
l[i] = 1
i -= 1
l[i] += 2
yield l
if __name__ == '__main__':
result = min(translate(l) for l in generator(13) if mult(l) > 4000000)
print("The result is:", result)
def listprimes(): for i in range(250): if (p.isprime(i)): print(i)
def test_seven_is_prime(self): self.assertTrue(prime.isprime(7))
def test_below2_notprime(self): for i in xrange(1, -10, -1): self.assertFalse(prime.isprime(i), msg = "Prime numbers must be greater than 1.")
def test_455_not_prime(self): self.assertFalse(prime.isprime(452))
from prime import is_prime_prob, isprime
import time
base = 2
exponent = 21701
starttime = time.time()
count = 0
try:
while True:
cur = time.time()
print(f"Checking {base}^{exponent}... ", end='')
if not isprime(exponent):
exponent += 2
print()
continue
if is_prime_prob(base**exponent - 1):
print(f"M({exponent}) is probably prime")
break
exponent += 2
count += 1
end = time.time() - cur
print(f"{end:.2f} seconds")
except KeyboardInterrupt:
print("Found {} probable primes in {} seconds".format(
count,
time.time() - starttime))
get_ipython().run_cell_magic('writefile', 'prime.py', "'''\nIt is a prime number\n'''\ndef isprime(num):\n '''\n It is\n '''\n if num in(0, 1):\n return False\n for prime in range(2, num-1):\n if num % prime == 0:\n return False\n return True")
# In[4]:
get_ipython().system(' pylint prime.py')
# In[5]:
import prime
prime.isprime(199)
# # USING UNIT TEST
# In[6]:
get_ipython().run_cell_magic('writefile', 'newprime.py', '\nimport unittest\nimport prime\n\nclass primenumber(unittest.TestCase):\n def testprime(self):\n Number = 32\n result = prime.isprime(Number)\n self.assertEquals(result, False)\n\n \n def testprimenum(self):\n Num = 199\n res = prime.isprime(Num)\n self.assertEquals(res, True)\n\n \n \nif __name__ == "__main__":\n unittest.main()')
# In[7]:
get_ipython().system(' python newprime.py')
def is_prime(n):
return prime.isprime(n)
import root
import prime
import key
li = []
print("give the range in which you want prime numbers")
print("start=", end=" ")
lower = int(input())
print("last=", end=" ")
print("should no exceed 1024", end=" ")
upper = int(input())
print("prime between", lower, "and", upper, "are:")
for num in range(lower, upper):
if (prime.isprime(num)):
print(num, end=" ")
li.append(num)
print()
print("select a prime", end=" ")
q = int(input())
if q not in li:
f1 = 1
while (f1):
if q not in li:
print("please select the prime number from below list only")
print(li)
q = int(input())
if q in li:
f1 = 0
print("primitive roots list")
li = []
print("enter lower and upper limits of primitive roots")
print("upper limit should not execeed " + str(q))
def is_8_prime_family(p, d):
c = 0
for r in '0123456789':
np = int(string.replace(p, d, r))
if(np > 100000 and np < 999999 and prime.isprime(np)): c += 1
return c==8
#!/usr/bin/env python
# -*- coding: utf-8 -*-
import prime
max_pair = (0, 0, 0)
for a in xrange(-999, 1000):
for b in xrange(max(2, 1 - a), 1000): # b >= 2, a + b + 1 >= 2
n, count = 0, 0
while True:
v = n * n + a * n + b
prime._refresh(v)
if prime.isprime(v):
count = count + 1
else:
break
n = n + 1
if count > max_pair[2]:
max_pair = (a, b, count)
print max_pair[0] * max_pair[1]
def is_truncatable_prime(n):
n_str = str(n)
if len(n_str) == 1: return False
leftwise = [int(n_str[:i]) for i in range(1, len(n_str))]
rightwise = [int(n_str[i:]) for i in range(len(n_str))]
return all(isprime(i) for i in chain(leftwise, rightwise))
def test_prime_positive():
assert prime.isprime(2) == True
assert prime.isprime(20) == False
assert prime.isprime(3)
assert not prime.isprime(4)
import prime print prime.prime(11) print prime._isprime(32) print prime.isprime(11)
def test_missing_param():
with pytest.raises(ValueError) as ex_info:
prime.isprime(-2)
assert ex_info.type == ValueError
def test_zeronot_prime(self): self.assertFalse(prime.isprime(0), msg = "Prime numbers must be greater than 1.")
def test_check_range_of_prime():
with pytest.raises(TypeError):
prime.isprime(False)
def test_five_is_prime(self): self.assertTrue(prime.isprime(5))
def test_check_prime_test():
assert prime.isprime(200) == False
It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 41^2 + 41 + 41 is clearly divisible by 41.
Using computers, the incredible formula n^2 - 79n + 1601 was discovered, which produces 80 primes for the consecutive values n = 0 to 79. The product of the coefficients, -79 and 1601, is -126479.
Considering quadratics of the form:
n^2 + an + b, where |a| <= 1000 and |b| <= 1000
where |n| is the modulus/absolute value of n
e.g. |11| = 11 and |4| = 4
Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0.
'''
import prime
max_pair = (0,0,0)
for a in xrange(-999, 1000):
for b in xrange(max(2, 1-a), 1000): # b >= 2, a + b + 1 >= 2
n, count = 0, 0
while True:
v = n*n + a*n + b
prime._refresh(v)
if prime.isprime(v): count = count + 1
else: break
n = n + 1
if count > max_pair[2]:
max_pair = (a,b,count)
print max_pair[0] * max_pair[1]
# -*- coding: utf-8 -*-
from __future__ import print_function, unicode_literals, absolute_import, division
"""
We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once.
For example, 2143 is a 4-digit pandigital and is also prime.
What is the largest n-digit pandigital prime that exists?
"""
digits = "123456789"
from prime import isprime
from itertools import permutations
from tuple_list_to_int_list import tuple_list_to_int_list
for i in range(9, 3, -1):
l = [
j for j in tuple_list_to_int_list(permutations(digits[:i]))
if isprime(j)
]
#print i, l
if any(l):
print("largest n-digit pandigital prime ", max(l))
break
