def main():
lst = array.array('Q', (1 for _ in range(LIMIT)))
lst[0] = 0
for i in range(2, LIMIT):
square = i * i
for j in range(i, LIMIT, i):
lst[j] += square
return sum(i for i, n in enumerate(lst) if euler.is_square(n))
def check_goldbach(n): prime_ptr = 0 while(True): prime = sieve[prime_ptr] if prime > n: break if euler.is_square((n - prime)/2): return True prime_ptr += 1 return False
def continued_fraction_period(n):
if is_square(n):
return 0
a_0 = floor(n ** 0.5)
m, d, a = 0, 1, a_0
ans = 0
while a != 2 * a_0:
m = d * a - m
d = (n - m ** 2) / d
a = floor((a_0+m) / d)
ans += 1
return ans
def continued_fraction_cycle(n):
if is_square(n):
return 0
a_0 = int(floor(n ** 0.5))
m, d, a = 0, 1, a_0
L = [a_0]
while a != 2 * a_0:
m = d * a - m
d = (n - m ** 2) / d
a = int(floor((a_0+m) / d))
L.append(a)
return L
def calculate(trio1, trio2):
d, e, a = trio1
f, c, a = trio2
if c < e:
d, e, f, c = f, c, d, e
b_2 = c * c - e * e
if euler.is_square(b_2):
x_2 = a * a + b_2
if x_2 % 2 == 0:
x = x_2 // 2
y = a * a - x
z = c * c - x
b = euler.int_sqrt(b_2)
x_y_z = int(x + y + z)
return x_y_z, min(a, b, c, d, e, f), max(a, b, c, d, e, f)
return None, None, None
def find_minimal_x(D):
if is_square(D):
return 0
a = continued_fraction_cycle(D)
h_1, h_2 = 1, a[0]
k_1, k_2 = 0, 1
a = a[1:]
i = 0
h_1, h_2 = h_2, a[i] * h_2 + h_1
k_1, k_2 = k_2, a[i] * k_2 + k_1
i = (i + 1) % len(a)
while h_2 ** 2 - D * k_2 ** 2 != 1:
h_1, h_2 = h_2, a[i] * h_2 + h_1
k_1, k_2 = k_2, a[i] * k_2 + k_1
i = (i + 1) % len(a)
return h_2