def test(p):
for i in xrange(len(p) - 1):
for j in xrange(i + 1, len(p)):
s1 = int(str(p[i]) + str(p[j]))
s2 = int(str(p[j]) + str(p[i]))
if not (is_prime(s1) and is_prime(s2)): return False
return True
def truncate_number(number):
if len(str(number)) >= 2:
right_num = number
len_rnum = len(str(right_num)) - 1
left_num = number
len_lnum = len(str(left_num))
primes_list = []
while len_rnum > 0:
right_num = str(right_num)[0:len_rnum]
right_num = int(right_num)
rnum_prime = common_functions.is_prime(right_num)
if rnum_prime == True:
#print("R:",str(right_num)[0:len_rnum], "is Prime")
primes_list.append(right_num)
#else:
#print("R:",str(right_num)[0:len_rnum], "is NOT Prime")
len_rnum -= 1
for y in range(1, len_lnum):
left_num = int(left_num)
new_l_num = str(left_num)[y:len_lnum]
new_l_num = int(new_l_num)
lnum_prime = common_functions.is_prime(new_l_num)
if lnum_prime == True:
#print("L:", str(left_num)[y:len_lnum], "is Prime")
primes_list.append(new_l_num)
#else:
#print("L:", str(left_num)[y:len_lnum], "is NOT Prime")
if len(str(number)) == 2:
if len(primes_list) == 2:
#print(primes_list)
truncatable_primes.append(number)
elif len(str(number)) == 3:
if len(primes_list) == 4:
#print(primes_list)
truncatable_primes.append(number)
elif len(str(number)) == 4:
if len(primes_list) == 6:
#print(primes_list)
truncatable_primes.append(number)
elif len(str(number)) == 6:
if len(primes_list) == 10:
#print(primes_list)
truncatable_primes.append(number)
def replace_digit(n):
'Returns True if number is part of an 8 prime value family'
string = str(n)
length = len(string)
digits = map(int, string)
unique = list(set(map(int, string)))
for d in unique:
count = 10
if not is_prime(n): count -= 1
for i in xrange(10):
if i != d:
new_digits = [
str(i) if ele == d else str(ele) for ele in digits
]
new = int(''.join(new_digits))
# disregard 0's in the beginning that shorten length
if new == 0 or math.floor(math.log10(new)) + 1 < length:
count -= 1
else:
if not is_prime(new): count -= 1
if count < 8: break
if count == 8: return True
return False
def num_of_circles(circles):
x = 8
current_num = 2
corners = 2
diagonals = [1]
for i in range(circles):
for z in range(current_num,current_num + x):
if z == (current_num + x - 1):
#print("UpperRight:",z-(corners*3))
#print("UpperLeft:",z-(corners*2))
#print("LowerLeft:",z-corners)
#print("LowerRight:",z)
diagonals.append(z-(corners*3))
diagonals.append(z-(corners*2))
diagonals.append(z-corners)
diagonals.append(z)
corners += 2
current_num += x
x += 8
num_of_primes_in_diagonals = 0
for diagonal in diagonals:
if common_functions.is_prime(diagonal) == True:
num_of_primes_in_diagonals += 1
print("\nNumber of Primes:",num_of_primes_in_diagonals)
print("Number of Diagonals:",len(diagonals))
percentage = num_of_primes_in_diagonals / len(diagonals) * 100
percentage = round(percentage, 10)
if num_of_primes_in_diagonals / len(diagonals) >= 0.1:
print("\nNumber of Primes is greater than 10%")
print(percentage,"%")
elif num_of_primes_in_diagonals / len(diagonals) < 0.1:
print("\nNumber of Primes is less than 10%")
print(percentage,"%")
print("\nLowerLeft:",z-corners)
print("LowerRight:",z)
print("Side Length:",(circles*2)+1)
def goldbachs_other_conjecture() -> int:
primes = [2]
squares = [2]
num_to_square = 1
num = 1
while True:
num += 2
# if num is prime number ---> add it to the primes list and continue
if is_prime(num):
primes.append(num)
continue
# if the last squares < num ---> increase num_to_square & add (2 * num_to_square^2) to squares
if squares[len(squares) - 1] < num:
num_to_square += 1
squares.append(2 * num_to_square ** 2)
# initiate primes_index to last item of primes & squares_index to 0
primes_index = len(primes) - 1
squares_index = 0
while True:
# run on squares until end or equation >= num
while squares_index < len(squares) and primes[primes_index] + squares[squares_index] < num:
squares_index += 1
# break the loop if found that the condition is not fulfilled
if primes[primes_index] + squares[squares_index] == num:
break
# start again with primes_index - 1
primes_index -= 1
squares_index = 0
# if not found primes_index - num found!
if primes_index == 0:
return num
prime_len = len(str_prime)
i = 0
# Generate the circular numbers
int_primes_list = []
while i < prime_len:
int_primes = int(str_prime)
int_primes_list.append(int_primes)
str_prime = str_prime[1:] + str_prime[0]
i += 1
#print(int_primes_list)
# Determine if all circular numbers are primes
are_all_circ_prime = all(
common_functions.is_prime(pr) is True for pr in int_primes_list)
if are_all_circ_prime is True:
for prime_val in int_primes_list:
if prime_val not in circular_primes:
circular_primes.append(prime_val)
num_of_cir_primes = len(circular_primes)
print(circular_primes)
print('Number of Circular Primes:')
print(num_of_cir_primes)
print('Duration: ')
print(datetime.now() - startTime)
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right
diagonal, but what is more interesting is that 8 out of the 13 numbers
lying along both diagonals are prime; that is, a ratio of 8/13 62%.
If one complete new layer is wrapped around the spiral above, a square
spiral with side length 9 will be formed. If this process is continued,
what is the side length of the square spiral for which the ratio of primes
along both diagonals first falls below 10%?
"""
from common_functions import is_prime
is_count = 8
not_count = 5
side, inc = 7, 8
n = 49
while float(is_count) / (is_count + not_count) > 0.1:
for i in xrange(4):
n += inc
if is_prime(n): is_count += 1
else: not_count += 1
side += 2
inc += 2
print side
# Project Euler - Problem #41 - Pandigital Prime - SOLVED - 62s
import common_functions
from datetime import datetime
startTime = datetime.now()
x = ''
pandigital_primes = []
for i in range(7000000 ,10000000):
prime = common_functions.is_prime(i)
if prime == True:
x = str(i)
chars = list(x)
max_char = max(chars)
if max_char == "2" and len(chars) == 2:
if "1" in chars:
pandigital_primes.append(i)
elif max_char == "3" and len(chars) == 3:
if "1" in chars and "2" in chars:
pandigital_primes.append(i)
elif max_char == "4" and len(chars) == 4:
if "1" in chars and "2" in chars and "3" in chars:
pandigital_primes.append(i)
elif max_char == "5" and len(chars) == 5:
if "1" in chars and "2" in chars and "3" in chars and "4" in chars:
pandigital_primes.append(i)
elif max_char == "6" and len(chars) == 6:
if "1" in chars and "2" in chars and "3" in chars and "4" in chars and "5" in chars:
# Project Euler - Problem #49 - Prime Permutations - Solved 0.02s
import common_functions
from datetime import datetime
startTime = datetime.now()
for x in range(1023, 9997):
a = common_functions.is_prime(x)
if a == True:
for y in range(3330, 3331):
second_sequence_num = x + y
b = common_functions.is_prime(second_sequence_num)
if b == True:
third_sequence_num = second_sequence_num + y
c = common_functions.is_prime(third_sequence_num)
if c == True:
first_sequence_str = str(x)
second_sequence_str = str(second_sequence_num)
third_sequence_str = str(third_sequence_num)
first_sequence_set = set(first_sequence_str)
second_sequence_set = set(second_sequence_str)
third_sequence_set = set(third_sequence_str)
final_str = first_sequence_str + second_sequence_str + third_sequence_str
final_str_set = set(final_str)
This is the longest sum of consecutive primes that adds to a prime below one-hundred. The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953. Which prime, below one-million, can be written as the sum of the most consecutive primes? """ from common_functions import prime_list from common_functions import is_prime max = 10**6 primes = prime_list(2, max) sums = [0] sum, idx = 0, 0 while (sum < max): sum += primes[idx] sums.append(sum) idx += 1 max_prime = 0 for i in range(idx): for j in range(i+1, idx): diff = sums[j] - sums[i] if is_prime(diff) and diff > max_prime: max_prime = diff print max_prime
else:
prime_sum += prime
prime_sum_list.append(prime_sum)
prime_dict = {prime_list[i]:prime_sum_list[i] for i in range(0,prime_list_len)}
'''
running_dict = {}
i = 0
while i < prime_list_len:
x = 1
running_sum = prime_list[i]
while x < prime_list_len:
if x > i:
running_sum += prime_list[x]
running_sum_prime = common_functions.is_prime(running_sum)
if running_sum_prime == True:
if running_sum < 1000000:
difference_len = (x+1)-i
running_dict[difference_len] = running_sum
x += 1
i += 1
print(max(running_dict.items()))
print('\nDuration: ')
print(datetime.now() - startTime)
def character_replacement(list_i):
for num_of_chars in range(1, len(list_i)):
print("\nNUM OF CHARS:", num_of_chars)
#print()
# Replace 1 Character
if num_of_chars == 1:
print("Replace 1 Character")
# START OF ORIGINAL
for x in range(0,len(list_i)):
val_primes_list = []
#print("NUM:",i)
#print("Char Count:",x)
for num in digits:
list_i[x] = str(num)
updated_str = "".join(list_i)
#print("List_I:", updated_str)
updated_num = int(updated_str)
if updated_num > 10 and updated_num < 100000:
if common_functions.is_prime(updated_num) == True:
val_primes_list.append(int(updated_str))
if i not in prime_digits:
prime_digits[i] = val_primes_list
#print(prime_digits)
elif i in prime_digits:
if len(val_primes_list) >= len(prime_digits[i]):
#print(val_primes_list)
#print(prime_digits[i])
prime_digits[i] = val_primes_list
#print(prime_digits)
list_i = list(str_i)
# END OF ORIGINAL
# Replace 2+ characters
elif num_of_chars == 2:
print("Replace 2 Characters")
for x in range(0,len(list_i)):
val_primes_list = []
print("NUM:",i)
print("Char Count:",x)
print("LEN LIST I:", len(list_i))
for num in digits:
#print("X:",x)
# First Number
if x == 0:
for num_vals in range(1,len(list_i)-1):
list_i[x] = str(num)
list_i[x+num_vals] = str(num)
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
elif x > 0 and x < len(list_i)-1:
num_difference = (len(list_i)-1) - 1
for num_vals in range(1,num_difference):
if x == 1:
list_i[x] = str(num)
list_i[x+num_vals] = str(num)
list_i[x] = str(num)
list_i[x+1] = str(num)
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
# Last Number
elif x == len(list_i)-1:
for num_vals in range(2,len(list_i)):
list_i[x] = str(num)
list_i[x-num_vals] = str(num)
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
if updated_num > 100 and updated_num < 100000:
if common_functions.is_prime(updated_num) == True:
val_primes_list.append(int(updated_str))
if i not in prime_digits:
prime_digits[i] = val_primes_list
print(prime_digits)
elif i in prime_digits:
if len(val_primes_list) >= len(prime_digits[i]):
print(val_primes_list)
print(prime_digits[i])
prime_digits[i] = val_primes_list
print(prime_digits)
list_i = list(str_i)
# START WORKING HERE - EVERYTHING ABOVE WORKS
elif num_of_chars == 3:
print("Replace 3 Characters")
for x in range(0,len(list_i)-1):
val_primes_list = []
print("NUM:",i)
print("Char Count:",x)
print("LEN LIST I:", len(list_i))
for num in digits:
#print("X:",x)
if x == 0:
for num_vals in range(1,len(list_i)-2):
list_i[x] = str(num)
list_i[x+num_vals] = str(num)
list_i[x+num_vals+1] = str(num)
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
elif x > 0 and x < len(list_i)-2:
num_difference = (len(list_i)-1) - 1
for num_vals in range(1,num_difference):
if x == 1:
list_i[x] = str(num)
list_i[x+num_vals] = str(num)
list_i[x+num_vals+1] = str(num)
else:
list_i[x] = str(num)
list_i[x+1] = str(num)
list_i[x+2] = str(num)
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
# Last Number
elif x == len(list_i)-1:
for num_vals in range(2,len(list_i)):
list_i[x] = str(num)
list_i[x-num_vals] = str(num)
list_i[x-num_vals] = str(num)-2
updated_str = "".join(list_i)
print("List_I:", updated_str)
updated_num = int(updated_str)
list_i = list(str_i)
if updated_num > 1000 and updated_num < 100000:
if common_functions.is_prime(updated_num) == True:
val_primes_list.append(int(updated_str))
if i not in prime_digits:
prime_digits[i] = val_primes_list
print(prime_digits)
elif i in prime_digits:
if len(val_primes_list) >= len(prime_digits[i]):
print(val_primes_list)
print(prime_digits[i])
prime_digits[i] = val_primes_list
print(prime_digits)
list_i = list(str_i)
if num_of_chars == 5:
print("Replace 5 Characters")
if num_of_chars == 6:
print("Replace 6 Characters") '''
digits = ['0', '1', '2', '3', '4', '5', '6', '7', '8', '9']
prime_digits = {}
for i in range(56003,56004):
str_i = str(i)
if common_functions.is_prime(i) == True:
list_i = list(str_i)
character_replacement(list_i)
max_key = max(prime_digits, key=lambda x: len(set(prime_digits[x])))
print("\nMAX KEY")
print(prime_digits[max_key])
print("Number of Primes:", len(prime_digits[max_key]))
print('\nDuration: ')
print(datetime.now() - startTime)
#print(primes_list)
truncatable_primes.append(number)
elif len(str(number)) == 4:
if len(primes_list) == 6:
#print(primes_list)
truncatable_primes.append(number)
elif len(str(number)) == 6:
if len(primes_list) == 10:
#print(primes_list)
truncatable_primes.append(number)
#else:
#print("Prime lengh == 1")
for i in range(22, 739398):
prime_number = common_functions.is_prime(i)
if prime_number == True:
#print(i)
truncate_number(i)
#print()
print(truncatable_primes)
truncatable_sum = sum(truncatable_primes)
print('# of Values:', len(truncatable_primes))
print("Total Sum:", truncatable_sum)
print()
print('Duration: ')
print(datetime.now() - startTime)