def diagonal_members(size):
diagonals = set()
for i in range(size, 0,-2):
# print("Layer ", i)
for j in range(0,4):
# print(i, j, i*i - (i-1)*j)
diagonals.add(i*i - (i-1)*j)
number_of_primes = sum(1 for x in diagonals if MathFunctions.miller_is_prime(x))
# print(number_of_primes,len(diagonals))
return (number_of_primes/len(diagonals))
'''
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?
'''
import MathFunctions, time
def is_pandigital(num):
str_rep = str(num)
string_rep_set = set(i for i in str_rep)
if str_rep.__contains__('0'):
return False
length = len(str_rep)
if length!= len(set(str_rep)):
return False
pan_digits = set(range(1,length+1))
num_digits = set(ord(i) - ord('0') for i in str_rep)
return num_digits.difference(pan_digits) == set()
start_time = time.time()
# cannot have 9 because 1+2+..+9 = 45, divisible always by 3 and 9, thus not a prime. Same for 8, sum is 36.
for i in range(7654321, 1, -2):
if is_pandigital(i) and MathFunctions.miller_is_prime(i):
print(i)
break
print("--- %s seconds ---" % (time.time() - start_time))
'''
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.
There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.
What 12-digit number do you form by concatenating the three terms in this sequence?
'''
import MathFunctions, itertools
def is_permuted(a, b):
list_of_permutations =[int(''.join(x)) for x in itertools.permutations(str(a))]
return list_of_permutations.__contains__(b)
for i in range(1489, 10000):
if MathFunctions.miller_is_prime(i) and MathFunctions.miller_is_prime(i+3330) and MathFunctions.miller_is_prime(i+6660)\
and is_permuted(i,i+3330) and is_permuted(i, i+6660):
print(i, i+3330, i+6660)
break
def sumOfPrimesBelow(upperLimit):
sum = 2
for i in range(3, upperLimit, 2):
if MathFunctions.miller_is_prime(i):
sum+=i
return sum
