def crt(a):
'''Returns the basic solution to the system x = a[i][0] (mod a[i][1]), i=0..len(a)-1 using the Chinese
Remainder Theorem. All a[i][1] must be co-prime (although a weaker condition involving the a[i][0]''s is
also sufficient for a solution, this implementation is not guaranteed to work for it).'''
a, n = zip(*a)
N = np.prod(map(long, n))
return sum(a[i] * (N / n[i]) * inv_mod(N / n[i], n[i]) for i in xrange(len(a))) % N
def Fnx(n, m, x):
'''Yields Fn(x) mod m for x,m s.t. gcd(1-x-x^2,m)=1.'''
F = Fib(r=m)
xn = pow(x, n, m)
return (inv_mod(1 - x - x * x, m) * ((x * (1 - F(n + 1) * xn - F(n) * xn * x)) % m)) % m
