The number of lattice points in a circle

Question

I am trying to determine the number of lattice points in a circle, i.e. basically I'm trying to find the number of pairs (m, n) such that m ^ 2 + n ^ 2 <= r ^ 2, where m and n are integers. I used the code below for this, but I get the wrong answer for r = 12, which should be 441 according to this , and I get 121, and I was wondering where I could be wrong:

def latticepoints(N):
    num = 0
    for m in range(0, int(N), 1):
        for n in range(0, int(N), 1):
            if (m**2 + n**2) <= N**2:
                num = num + 1

    return number

print(latticepoints(12))

Edit

Just decided. Just need to change the loops to:

for m in range(-int(N), int(N)+1, 1):
    for n in range(-int(N), int(N)+1, 1):

thanks

Answer 1

As you already noted, the problem is that you count the lattice points in one quadrant of the circle. Expanding the range for correction is one approach; an alternative solution is to take

lattice points = 4 * (lattice points in a single quadrant) - 3

3, .

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