Thread: no positive integers

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.

Originally Posted by Fayyaz

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.

Hi

m^2 + 1)" alt="m^4 + 2m^3 + 2m^2 + 2m + 1 = m^4 + 2m^2 + 1 + 2m^3 + 2m
= (m^2 + 1)^2 + 2m\m^2 + 1)" />

m^2 + 2m + 1) = (m^2 + 1)\m+1)^2" alt="m^4 + 2m^3 + 2m^2 + 2m + 1 = (m^2 + 1)\m^2 + 2m + 1) = (m^2 + 1)\m+1)^2" />

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.

thanks guys

how should i proceed with the next bit:

Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?

Hello Fayyaz

Originally Posted by Fayyaz

thanks guys

how should i proceed with the next bit:

Are there any positive integers m for which m^4 +m^3 +m^2 +m+1 is a perfect square?

When . So there's certainly one value. Whether there are any more, I don't know.

Grandad

hi

thanks, yeah i figured that out using excel spreadsheet, but anyway I can do an algebraic proof?