Proving that an expression is a perfect square for a finite number of values of n

**Aquafina** · May 15th 2009, 08:24 AM

Prove that

is the only integer making

a perfect square

**TwistedOne151** · May 15th 2009, 09:37 AM

What about n=0 and n=-1?

--Kevin C.

**Aquafina** · May 15th 2009, 10:11 AM

True. Ok for 0 and 3 then..

**Media_Man** · May 15th 2009, 01:29 PM

Conjecture: The only integer solutions to the function $f(x)=\sqrt{x^4+x^3+x^2+x+1}$ are $(-1,1)$ , $(0,1)$ , and $(3,11)$ . No others exist. Furthermore, $\lim_{|n|\rightarrow \infty} frac(f(n))=\frac{7}{8}$ for n odd and $\frac{3}{8}$ for n even, when $n \in \mathbb{Z}$ .

I agree, it is an interesting problem, and looking at it numerically, I believe it to be true. But why these three solutions should exist uniquely, I do not know.

**Media_Man** · May 15th 2009, 02:53 PM

UPDATE: http://www.mathhelpforum.com/math-he...duction-3.html Proof for general case found a posteriori.

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Proving that an expression is a perfect square for a finite number of values of n

Interesting problem indeed

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