Re: ranges of integer polynomials

On Sun, 07 Oct 2007 01:02:19 -0400, quasi <quasi@xxxxxxxx> wrote:

Just to give an update ...

Of the 12 problems I posed in this thread, 3 of them -- problems 1, 6,
8, have now been resolved.

There are 9 problems still left from the original group, and I believe
that they are all within reach.

For reference, here are the problems not yet resolved:

Let f be an integer polynomial, possibly multivariate, and let
range(f) denote the range of f for all integer inputs.

problem (2):

Can range(f) = {x^2 | x in Z} union {-x^2 | x in Z} ?

problem (3):

(a) Can range(f) = {x^2 | x in Z} union {2x | x in Z} ?

(b) Can range(f) = {x^2 | x in Z} union {2x | x in N} ?

problem (4):

(a) Can range(f) = {x^2 | x in Z} union {x^3 | x in Z} ?

(b) Can range(f) = {x^2 | x in Z} union {x^3 | x in N} ?

problem (5):

Must at least one of the sets

(range(f) intersect N)

(range(-f) intersect N)

have a density, as a subset of N?

problem (7):

If range(f) is a subset of N, must range(f) have a density, as a
subset of N?

problem (9):

(a) Can range(f) be the set of all integer non-cubes?

(b) Can range(f) be the set of all positive integer non-cubes?

problem (10):

(a) For which positive integers k>2, if any, can the set of all
integer non-k'th powers be realized as range(f)?

(b) For which positive integers k>2, if any, can the set of all
positive integer non-k'th powers be realized as range(f)?

problem (11):

Can range(f) = the set of squarefree positive integers?

problem (12):

(a) If f is not constant and range(f) contains zero as a least
element, must there exist an integer polynomial g such that range(g) =
range(f) \ {0} ?

(b) If f is not constant and range(f) is a subset of N, must there
exist an integer polynomial g such that range(g) = range(f) union {0}
?

I now have solutions for problems 2, 3, 4.

The outcome was what I expected -- for all 3 of them, the answer is
"no".

I'll post proofs when I get a chance.

quasi
.

Relevant Pages

  • Re: ranges of integer polynomials
    ... On the complement of the k-th powers for fixed k>1, a ... powers is the range of a polynomial. ... A Polynomial whose Range is the Positive Integers which are not ... It is convenient to have all variables range over the non-negative ...
    (sci.math)
  • REPOST: Re: Pomerance/Crandall "Prime Numbers" book: inequality question
    ... >which I already own (I bought it a few years ago and only read a few ... >Why is this inequality true? ... one of those powers of p for each prime p <= x and multiplying them ... This gives us Pdistinct positive integers, ...
    (sci.crypt)
  • Re: Pomerance/Crandall "Prime Numbers" book: inequality question
    ... >out, and before I buy it I figured I should read the first edition, ... >which I already own (I bought it a few years ago and only read a few ... one of those powers of p for each prime p <= x and multiplying them ... This gives us Pdistinct positive integers, ...
    (sci.crypt)
  • Re: ranges of integer polynomials
    ... Can rangebe the set of all positive integer non-cubes? ... For which positive integers k>2, if any, can the set of all ... positive integer non-k'th powers be realized as range? ...
    (sci.math)
  • Re: ranges of integer polynomials
    ... try whichever one appeals to them. ... Can rangebe the set of all positive integer non-cubes? ... For which positive integers k>2, if any, can the set of all ... positive integer non-k'th powers be realized as range? ...
    (sci.math)