Show that the product of two of the numbers $(65^{1000} - 8^{2001} + 3^{177}), (79^{1212} - 9^{2399} + 2^{2001})$ and $(24^{4493} - 5^{8192} + 7^{1777})$ is non-negative, without actually evaluating the numbers.
P.S. I have found by calculation that all the three numbers are positive, but that does not solve the problem of proving without calculation.
Thanks in advance.