ranges of integer polynomials

Listed below are some previously posted but unresolved problems
relating to this topic, plus a few new ones ...

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

problem (1): [tommy1729's problem]

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

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 (7):

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

problem (8):

If range(f) is a subset of N, must range(f) be recursive?

quasi
.