Perfect Square or not?

Is the number N = 1977 · 1978 · 1979 · 1979 + 1 a perfect square?

Answer

Let 1977 = X.

Then N = X(X + 1)(X + 2)(X + 3) + 1 = (X2 + 3X)(X2 + 3X + 2) + 1.

Let X2 + 3X = y.

Then N = Y(Y + 2) + 1 = Y2 + 2Y + 1 = (Y + 1)2 = (X2 + 3X + 1)2 = (19772 + 3×1977 + 1)2.

Yes, N is a perfect square.

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