moved to a new thread for general case
I have not had luck with induction here. However, here are two very unusual observations I have made:
for n even and for n odd. In other words, the decimal expansion seems to be tending towards an alternating pattern between .375 and .875. To be a perfect square, this decimal would of course need to be exactly zero, so proving this observation would imply your hypothesis.
Since it does alternate, you may try using induction where
And don't forget the identity
So far as I know, induction is not effective for proving a limit exists. Here is what I have gotten so far:
Let
As n gets large,
More rigorously,
The actual value is not so important as the fact that it is an integer. What this essentially means is that if is not a perfect square, then won't be either, since both of their square roots will have increasingly identical decimal expansions or .
You can actually graph and and see how astonishingly closely they overlap. I am currently looking around for a method of taking the square root of a polynomial. I'm pretty sure it exists, basically given , find coefficients that satisfy , effectively finding the square root of a polynomial.
I believe that rewriting as series, a bunch of stuff will cancel, leaving only and some extra term that goes to zero fairly quickly.
I have the sneaking suspicion that I am using a cannon to kill a mosquito, and a super member will come along with a painfully obvious elementary proof.
yeah but the original question was to find any values of n for which
n^4+n^3+n^2+n+1 is a perfect square
i found out that it works for n=3, and guessed that there would be no other value, using excel for very large value of n!
so i decided to prove
(n^5 - 1)/(n-1) ≠ k^2 for n>3
The original question, as far as I am concerned is:
Hi, I need to show that for n > 3:
(n^5 - 1)/(n-1) ≠ k^2 (does not equal)
for any integer 'k'
i.e. it is not a perfect square
And of course my argument only holds for the proof you wanted.(make that "initially" wanted )