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.
$\displaystyle N \;=\;m^4 + 2m^3 + 2m^2 + 2m + 1 \;=\; m^4 + 2m^2 + 1 + 2m^3 + 2m$
. . $\displaystyle =\; (m^2 + 1)^2 + 2m\,(m^2 + 1) \;=\; (m^2 + 1)\,(m^2 + 2m + 1) \;=\; (m^2 + 1)\,(m+1)^2$
If $\displaystyle N$ is a square, then $\displaystyle (m^2+1)$ must be a square.
. . Hence, $\displaystyle m^2 + 1 \:=\:a^2\quad\hdots$ for some integer $\displaystyle a.$
Then: .$\displaystyle a^2-m^2 \:=\:1 \quad\hdots $ two squares differ by 1.
. . But the only two squares that differ by 1 are: .$\displaystyle a = \pm1,\text{ and }\, m = 0$
Since $\displaystyle m$ must be a positive integer, no solution exists.