Prove that for no integer n, n^6 +3n^5 -5n^4 -15n^3 +4n^2 +12n +3 is a perfect square.

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Solution

$Given, n^{6} + 3 n^{5} - 5 n^{4} - 15 n^{3} + 4 n^{2} + 12 n + 3 = n^{5} (n + 3) - 5 n^{3} (n + 3) + 4 n (n + 3) + 3 = (n + 3) (n^{5} - 5 n^{3} + 4 n) + 3 = n (n + 3) (n^{4} - 5 n^{2} + 4) + 3 = n (n + 3) (n^{4} - 4 n^{2} - n^{2} + 4) + 3 = n (n + 3) (n^{2} (n^{2} - 4) - 1 (n^{2} - 4)) + 3 = n (n + 3) ((n^{2} - 4) (n^{2} - 1)) + 3 = n (n + 3) (n - 1) (n + 1) (n - 2) (n + 2) + 3 Since we can see that above cannot be written for any form of a perfect square therefore it is not a perfect square .$

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