def right_trunc(num):
while num != 0:
if check_prime(num):
num = (num - (num % 10)) / 10
else:
return False
return True
def left_trunc(num):
i = 1
temp = 0
while (temp != num):
temp = num % (10**i)
if check_prime(temp):
i = i + 1
else:
return False
return True
def distinct_factors(num):
if check_prime(num):
return 1
number = copy.copy(num)
i = 3
count = 0
if number % 2 == 0:
while number % 2 == 0:
number = number / 2
count = 1
while (count <= 4 and number != 1):
if number % i == 0:
if check_prime(i):
count = count + 1
while number % i == 0:
number = number / i
i = i + 2
return count
list_h.remove(
h)
if (len(list_h)
!= 0):
for i in list_h:
if (check_prime(
int(
a
+
b
+
c
+
d
+
e
+
f
+
g
+
h
+
i
)
)):
pan_prime.append(
int(
a
+
b
# 15 = 7 + 2×2^2
# 21 = 3 + 2×3^2
# 25 = 7 + 2×3^2
# 27 = 19 + 2×2^2
# 33 = 31 + 2×1^2
#
# It turns out that the conjecture was false.
#
# What is the smallest odd composite that cannot be written as the sum of a prime and twice a square?
from math import *
from common_functions import check_prime, timing
timer = timing()
flag = True
odd_comp = 35
while (flag):
if (not(check_prime(odd_comp))):
GoldBach = False
for i in range (3, odd_comp, 2):
if (check_prime(i)):
if (sqrt((odd_comp - i) / 2) == int(sqrt((odd_comp - i) / 2))):
GoldBach = True
if (not GoldBach):
flag = False
odd_comp = odd_comp + 2
timer("The smallest odd number that does not satisfy Goldbach's conjecture is {}".format(odd_comp - 2))
# -*- coding: utf-8 -*-
# What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20 ?
from common_functions import check_prime, timing
def highest_power_less_than_number(prime, number):
a = 1
while(a <= number):
a = a * prime
return a/prime
timer = timing()
i = 2
ans = 1
number = 20
while (i <= number):
is_prime = check_prime(i)
if (is_prime): # All primes greater than 5 have only these numbers in decimal place
ans = ans * highest_power_less_than_number(i, number)
i = i + 1
timer("The smallest positive number that is evenly divisible by all of the numbers from 1 to 20 is {}".format(ans))
# n2+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.
from math import *
from common_functions import timing, check_prime, PrimeSieve
timer = timing()
b = PrimeSieve(1000)
a_b = []
for i in range(-999, 1000, 1):
for j in b:
if (((i + j + 1) > 0) and (check_prime(i + j + 1))):
a_b.append([i, j])
n = 0
while (len(a_b) != 1):
for ab in a_b[:]:
if len(a_b) != 1:
if (((n**2) + (ab[0] * n) + ab[1]) <= 0):
a_b.remove(ab)
continue
val = check_prime(((n**2) + (ab[0] * n) + ab[1]))
if (val != 1):
a_b.remove(ab)
n = n + 1
timer("The product of the coefficients is {}".format(a_b[0][0] * a_b[0][1]))
if ((num % 2) == 0):
return False
else:
num = (num - (num % 10)) / 10
return True
def rotate_num(num):
last_digit = num % 10
num = (num - (last_digit)) / 10
return (int(float(str(last_digit) + str(num))))
timer = timing()
count = 13
for i in range(101, 1000000, 2):
if check_any_even_number(i):
if (check_prime(i)):
flag = 1
l = len(str(i))
for j in range (0, l):
i = rotate_num(i)
if (check_prime(i)):
continue
else:
flag = 0
break
if flag == 1:
count = count + 1
timer("There are {} circular primes below one million".format(count))