Proof by Induction?

Aquafina · 2009-05-15 02:31:18

Hi I want to show that for x>3

(x^5 - 1)/(x-1) ≠ y^2

where y is any integer

basically the expression is not a perfect square

I've tried proof by induction but can't seem to get anywhere

TheDude · 2009-05-20 00:29:24

Are you sure you're supposed to use induction? It may be possible, but don't see a good way of solving the problem with it. I would go for a more direct approach.

First, let's simplify the expression. The LHS is equal to

for all x other than x=1. Since we're only interested in x>3, we can use this simplification.

Let's assume that this expression is a perfect square. That means we can factor it as follows:

Solve for a, b, c:

Kurre · 2009-05-20 05:43:20

no, a value for a polynomial can be a perfect square, without the polynomial being a perfect square of a polynomial...

bobbym · 2009-05-20 11:52:04

Hi Kurre;

I'm not following you. When x=3, f(3) = 121 is a square.

Last edited by bobbym (2009-05-20 17:13:04)

Avon · 2009-05-20 12:58:24

bobbym wrote:

I'm not following you. When x=3, f(3) = 121 a square.

This exactly illustrates Kurre's point. TheDude's post presents a proof that the polynomial

is not the square of a polynomial.
Nevertheless, is a square.

bobbym · 2009-05-20 17:16:39

Sorry Kurre and thanks Avon;

I get it now. So, the original question still remains unproved?

Math Is Fun Forum

#1 2009-05-15 02:31:18

Proof by Induction?

#2 2009-05-20 00:29:24

Re: Proof by Induction?

#3 2009-05-20 05:43:20

Re: Proof by Induction?

#4 2009-05-20 11:52:04

Re: Proof by Induction?

#5 2009-05-20 12:58:24

Re: Proof by Induction?

#6 2009-05-20 17:16:39

Re: Proof by Induction?

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