def check_prime_concatenations_mr(prime1, prime2):
str_prime1 = str(prime1)
str_prime2 = str(prime2)
status = True
prime_cat = int(str_prime1 + str_prime2)
if not common.is_prime_mr(prime_cat):
status = False
prime_cat = int(str_prime2 + str_prime1)
if not common.is_prime_mr(prime_cat):
status = False
return status
def count_reduced_fractions(max_d):
count = 0
# Even denominators
for d in range(2, max_d+1, 2):
count += 1 # always count 1/d.
for n in range(3, d, 2): # Check only odd numerators
if common.get_gcd(n, d) == 1:
count += 1
# Odd denominators
for d in range(3, max_d+1, 2):
if common.is_prime_mr(d):
count += d - 1
else:
count += 2 # always count 1/d and 2/d for odd d.
for n in range(3, d):
if common.get_gcd(n, d) == 1:
count += 1
return count
def generate_totient_list(max_n):
"""
totient(m*n) = totient(m)*totient(n)*(d/totient(d)) where d = gcd(m, n)
totient(2*m) = 2*totient(m) if m is even
= totient(m) if m is odd
totient(n^m) == n^(m-1) * totient(n)
:param max_n:
:return:
"""
totient_list = [0] * (max_n+1)
prime_list = [2]
# Process 2 and multiples of 2 first
n = 2
totient_list[n] = n - 1
prev_phi = totient_list[n]
n2 = 2 * n
while n2 < max_n:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
# Loop through rest of numbers
for n in range(3, max_n+1):
if totient_list[n] == 0:
if common.is_prime_mr(n): # all primes > 2 are odd
totient_list[n] = n - 1
n2 = 2 * n
if n2 < max_n:
totient_list[n2] = totient_list[n]
prev_phi = totient_list[n]
n2 *= 2
while n2 < max_n:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
# for p in prime_list:
# index = p * n
# if index < max_n:
# totient_list[index] = totient_list[p] * totient_list[n]
# else:
# break
# prime_list.append(n)
m = 2
index_prev = n**(m - 1)
index = n**m
while index < max_n:
if totient_list[index] == 0:
totient_list[index] = index_prev * totient_list[n]
m += 1
index_prev = index
index = n**m
for m in range(3, n):
index = n * m
if index < max_n:
if totient_list[index] == 0:
totient_list[index] = totient_list[m] * totient_list[n]
else:
break
else: # n is not prime
totient_list[n] = common.totient(n)
prev_phi = totient_list[n]
n2 = 2 * n
if (n % 2 == 0) and (n2 < max_n): # if n is odd
if totient_list[n2] == 0:
totient_list[n2] = prev_phi
n2 *= 2
while n2 < max_n:
if totient_list[n2] == 0:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
else:
prev_phi = totient_list[n]
n2 = 2 * n
if (n % 2 == 1) and (n2 < max_n): # if n is odd
if totient_list[n2] == 0:
totient_list[n2] = prev_phi
n2 *= 2
while n2 < max_n:
if totient_list[n2] == 0:
totient_list[n2] = 2 * prev_phi
prev_phi = totient_list[n2]
n2 *= 2
return totient_list