def main(verbose=False):
prime_max = 10 ** 5
PRIMES = sieve(prime_max)
found = []
n = 2
while len(found) < 25:
n += 1
if n > prime_max:
prime_max *= 10
PRIMES = sieve(prime_max)
if n % 2 == 0 or n % 5 == 0 or n in PRIMES:
continue
basis = n
if n % 3 == 0:
basis = 9 * n
if (n - 1) % order_mod_n(10, basis) == 0:
found.append(n)
if verbose:
return ('%s.\nAs a check, the first five values are calculated to be '
'%s, as stated.' % (sum(found),
', '.join(str(num) for num in found[:5])))
else:
return sum(found)
def hyper_exponentiate(a, b, modulus):
if modulus == 1:
return 1
elif modulus == 2:
return a % 2
if b == 1:
return a % modulus
a_order = order_mod_n(a, modulus)
return (a ** hyper_exponentiate(a, b - 1, a_order)) % modulus
def main(verbose=False):
max_a = 0
n = 10 ** 6
while max_a <= 10 ** 6:
n += 1
if gcd(10, n) == 1:
basis = n
if n % 3 == 0:
basis = 9 * n
curr_order = order_mod_n(10, basis)
if curr_order > max_a:
max_a = curr_order
return n
def main(verbose=False):
# We need to find the residue of P modulo 5**10
# since we already know the residue modulo 2**10
actual_exponent = 7830457 % order_mod_n(2, 5 ** 10)
# want to find 2 raised to this exponent
power_of_two = 1
for i in range(actual_exponent):
power_of_two = (2 * power_of_two) % 5 ** 10
residue = (28433 * power_of_two + 1) % 5 ** 10
unit_two_null_five = unit_a_null_b(2 ** 10, 5 ** 10)
unit_five_null_two = unit_a_null_b(5 ** 10, 2 ** 10)
return (1 * unit_two_null_five + residue * unit_five_null_two) % 10 ** 10
def main(verbose=False):
max_index = -1
max_block_size = -1
for i in range(1, 1000):
stripped_val = robust_divide(robust_divide(i, 2), 5)
if stripped_val == 1:
block_size = 0
else:
block_size = order_mod_n(10, stripped_val)
if block_size > max_block_size:
max_block_size = block_size
max_index = i
return max_index
