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
def is_squarefree(n):
if n == 1:
return True
quotient = n
count = 1
while count == 1:
prime, _ = first_prime_divisor(quotient)
quotient, count = robust_divide(quotient, prime, include_count=True)
if quotient == 1:
return (count == 1)
return False
def last5(n):
if n < 8:
# The remaining part is always divisible by
# 2**5 for n > 7, these are special cases
solutions = {0: 1,
1: 1,
2: 2,
3: 6,
4: 24,
5: 12,
6: 72,
7: 504}
return solutions[n]
if n <= 5 ** 5:
residues = {}
for i in range(1, n + 1):
to_add = robust_divide(i, 5)
# if to_add is not in residues, sets to 1
# (default 0 returned by get)
residues[to_add] = residues.get(to_add, 0) + 1
else:
residues = {}
for residue in range(1, 5 ** 5):
if residue % 5 != 0:
residues[residue] = (n - residue) / (5 ** 5) + 1
max_power = int(floor(log(n) / log(5)))
for power in range(1, max_power + 1):
biggest_quotient = n / (5 ** power)
for residue in range(1, 5 ** 5):
if residue % 5 != 0:
residues[residue] += ((biggest_quotient - residue) /
(5 ** 5) + 1)
product = 1
for residue, power in residues.items():
power_apply = power % (4 * (5 ** 4)) # PHI(5**5)
product = (product * (residue ** power_apply)) % (5 ** 5)
fives = num_factors_fact(n, 5) % (4 * (5 ** 4)) # PHI(5**5)
inverse = inverse_mod_n(2, 5 ** 5)
product = (product * (inverse ** fives)) % (5 ** 5)
return (product * unit_a_zero_b(5 ** 5, 2 ** 5)) % 10 ** 5
def k_reduced(N):
k_st = k_star(N)
k_induced = robust_divide(robust_divide(k_st, 5), 2)
shared_factors = gcd(k_induced, N)
return k_induced / shared_factors
