complement of the range of an integer polynomial
- From: quasi <quasi@xxxxxxxx>
- Date: Wed, 26 Sep 2007 18:21:36 -0400
tommy1729 asked whether or not there is an integer polynomial whose
range, for integer inputs, is the set of all integers except for the
perfect squares.
Alternatively, one could ask -- does there exist an integer polynomial
whose range, for integer inputs, contains all nonnegative integers
except for the perfect squares.
For either of the above questions, if there is such a polynomial, it's
clear that it can't be univariate.
I don't recall whether tommy conjectured for existence or
non-existence, but I conjectured for non-existence.
Neither question has yet to be answered in sci.math.
Perhaps a slightly more general viewpoint will help resolve some of
these questions. Thus, consider the following generalization ...
Let f be an integer polynomial and let range(f) denote the range of f
for all possible integer inputs.
Conjecture:
If N\range(f) is infinite, then it has positive density as a subset of
N.
quasi
.
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