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euler621.py
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euler621.py
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from eulertools import primes, isqrt
def is_square(n, p):
return pow(n, (p-1) // 2, p) == 1
def exp_in_fp2(l, exponent, k, p):
if exponent == 0:
return (1, 0)
elif exponent == 1:
return l
elif exponent % 2 == 0:
x, y = l
return exp_in_fp2(((x**2+y**2*k) % p , (2*x*y) % p), exponent//2, k, p)
else:
x, y = l
z, w = exp_in_fp2(((x**2+y**2*k) % p , (2*x*y) % p), (exponent-1)//2, k, p)
return ((x*z+y*w*k) % p, (x*w+y*z) % p)
def find_square_roots(n, p):
if n == 1:
return 1, p-1
"""Implementing Cipolli's algorithm"""
a = 1
while is_square((a**2-n) % p, p):
a += 1
result = exp_in_fp2((a, 1), (p+1)//2, (a**2-n) % p, p)
return result[0], p - result[0]
def power(p, n):
count = 0
while n % p == 0:
n //= p
count += 1
return count
def G(n):
count = 0
prime_list = primes(isqrt(4*n+1)+1)
N = 8*n+3
l = [1 for _ in xrange((isqrt(8*n+3)-1)//2+1)]
products = [2 for _ in xrange((isqrt(8*n+3)-1)//2+1)]
for p in prime_list[1:]:
L = N % p
if L == 0:
sol = ((-1)*(p+1)//2) % p
for i in xrange(sol, len(l), p):
exp = power(p, N-(2*i+1)**2)
products[i] *= p**exp
if p % 4 == 1:
l[i]*= (exp+1)
else:
if exp % 2 != 0:
l[i] = 0
if is_square(L, p):
l1, l2 = find_square_roots(L, p)
sol1 = ((-1 + l1)*(p+1)//2) % p
sol2 = ((-1 + l2)*(p+1)//2) % p
for i in range(sol1, len(l), p) + range(sol2, len(l), p):
exp = power(p, N-(2*i+1)**2)
products[i] *= p**exp
if p % 4 == 1:
l[i]*= (exp+1)
else:
if exp % 2 != 0:
l[i] = 0
for k in xrange((isqrt(8*n+3)-1)//2+1):
if N - (2*k+1)**2 != products[k]:
p = (N - (2*k+1)**2)//products[k]
if p % 4 == 3:
l[k] = 0
else:
l[k] *= 2
return sum(l)
print G(17526*10**9)