Re: ranges of integer polynomials

Here's the latest update ...

Of the 13 problems I posed in this thread, some are now solved ...

Adler posted solutions to problems (1), (7), (8).

I posted a solution to problem (6), but the solution was invalid, so
that problem is still unresolved.

It seems clear that some modification of the method adler used to
solve problem (8) will probably also work to clinch problem (6).

Similarly, some modification of the method adler used to solve problem
(7) will probably also work to clinch problem (5).

I now have solutions to problems 2, 3, 4, but I won't have time to
post them until probably Tuesday, so I'll leave them in the list for
now.

Thus, problems 1,7,8 are solved, problem 2,3,4 are claimed as solved,
and problem 5,7 are likely to be quickly solved.

The possibly harder problems are problems 9 through 13.

Here's the list, omitting problems 1,7.8 ...

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 (6):

Must at least one of the sets

(range(f) intersect N)

(range(-f) intersect N)

be recursive?

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}
?

problem (13): [tommy1729's generalized "deny f" problem]

If range(f) is a subset of N and also recursive, must N\range(f) be
the range of an integer polynomial?

quasi
.

Relevant Pages