Show no positive integers m for which m^4 + 2m^3 + 2m^2 + 2m + 1 is a perfect square. Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?
If so, find all such m.
Hello, Fayyaz!
running-gag has an excellent proof, but omitted the punchline.
. .
If is a square, then must be a square.
. . Hence, for some integer
Then: . two squares differ by 1.
. . But the only two squares that differ by 1 are: .
Since must be a positive integer, no solution exists.