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Topic: Square roots of polynomials
Replies: 3   Last Post: Mar 25, 2014 4:59 AM

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Jose Carlos Santos

Posts: 4,872
Registered: 12/4/04
Re: Square roots of polynomials
Posted: Mar 25, 2014 4:59 AM
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On 25-03-2014 7:00, quasi wrote:

>> Let P(x) be a polynomial with integer coefficients such that,
>> for each natural number _n_, P(n) is a perfect square. Does it
>> follow that P(x) is the square of some polynomial with integer
>> coefficients?

>
> Yes.
>
> My proof hinges on a key result which I call "The Sandwich
> Theorem". It's an elementary result, definitely known, but
> perhaps not well enough known.
>
> I independently discovered and proved this result when I was
> in high school.
>
> I'll state the theorem and some corollaries, without proof for
> now, but I'll supply proofs if requested.


Cute! Thanks a lot.

Best regards,

Jose Carlos Santos




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