from math import gcd


def euler_139():
    limit = 100_000_000
    r, m, n = 0, 2, 1
    while True:
        a = m * m - 1 * 1
        b = 2 * m * 1
        c = m * m + 1 * 1
        if a + b + c > limit:
            return r

        for n in range(1, m):
            if (m - n) % 2 == 1 and gcd(m, n) == 1:  # Ensure m and n are coprime and not both odd
                a = m * m - n * n
                b = 2 * m * n
                c = m * m + n * n
                if a > b:
                    a, b = b, a

                ka, kb, kc = a, b, c 
                while ka + kb + kc < limit:
                    assert ka * ka + kb * kb == kc * kc
                    if kc % (kb - ka) == 0:
                        r += 1
                    ka, kb, kc = ka + a, kb + b, kc + c
        m += 1


if __name__ == "__main__":
    solution = euler_139()
    print("e139.py: " + str(solution))
    assert solution == 10057761