Let m and n be positive integers, and prove that y = 1 / 2 [M ^ 4 + n ^ 4 + (M + n) ^ 4] is a complete square number

Let m and n be positive integers, and prove that y = 1 / 2 [M ^ 4 + n ^ 4 + (M + n) ^ 4] is a complete square number

y＝（1/2）［m^4＋n^4＋（m＋n）^4］
＝（1/2）［（m^4＋2（mn）^2＋n^2）－2（mn）^2＋（m^2＋n^2＋2mn）^2］
＝（1/2）（m^2＋n^2）^2－（mn）^2＋（1/2）（m^2＋n^2）＋2（mn）^2＋2mn（m^2＋n^2）
＝（m^2＋n^2）^2＋2mn（m^2＋n^2）＋（mn）^2
＝［（m^2＋n^2）＋mn］^2.
∵ m and N are integers, and∵ y is a complete square

Given that 10 ^ m = 5 and 10 ^ n = 6, find the value of 10 ^ 3M + 10 ^ 3N?

10^3m+10^3n
=(10^m)^3+(10^n)^3
=5^3+6^3
=125+216
=341

Given m / N = 5 / 3, find the value of (1 / M + N + 1 / m-n) △ 1 / N-N / M-N △ m + n / n,
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From M / N = 5 / 3, we obtain the original formula of M = 5N / 3 = {(m-n) / (m-n) * (M + n) + (M + n) / (m-n) * (M + n)} * N-N square / (m-n) * (M + n) = 2Mn / (m-n Square) - n square / (m-n Square) = (2mn-n Square) / (m-n Square) and substitute M = 5N / 3 into = (7n square / 3) * (9 △ 16N Square) = 21 / 16