Re: polynomials whose values are all perfect squares

On Fri, 21 Sep 2007 21:21:03 -0400, quasi <quasi@xxxxxxxx> wrote:

On Fri, 21 Sep 2007 20:06:10 -0500, Robert Israel
<israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:

quasi <quasi@xxxxxxxx> writes:

tommy1729 wrote:

quasi wrote:

Let f be an n-variate polynomial with integer coefficients,
regarded as a function from Z^n to Z. Suppose every element of
f(Z^n) is the square of an integer. Must f = g^2 for some n-variate
polynomial g with integer coefficients?

i asked a similar question about a month ago.

Actually, if I recall correctly, you asked if there was an integer
polynomial whose range was all non-squares. A good question and, as
of now, still unanswered.

Really? But that's too easy. Try x^2 + 2 or -1 - x^2.

No -- the range is required to be _all_ non-squares.

Clearly, a counterexample, if any, will need to be multivariate.

I see the source of the confusion ...

The non-squares range problem was from a different thread, and in that
thread, the interpretation of "all" was stronger.

Here is tommy1729's question, stated more precisely ...

Let S = { x^2 | x in Z }.

Does there exist an n-variate polynomial f such that f(Z^n) = Z\S ?

I believe tommy predicted "yes", whereas I predicted "no".

The problem is still unresolved.

quasi
.

Relevant Pages
Quantcast