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It is a long standing problem to investigate whether irreducible integral polynomials not divisible by a fixed square integer assumes square-free values infinitely often. The result is known conditioned on the $abc$-conjecture (due to Granville), and the best unconditional results are for $k$-free values of polynomials. That is, one can prove that an irreducible polynomial with no fixed $k$-th power divisor of degree $d$ assumes infinitely many $k$-free values if $k$ is sufficiently large with respect to $d$. For the single variable case, it is known that an irreducible polynomial $f(x)$ of degree $d$ with no fixed $k$-th power divisor assumes infinitely many $k$-free values as soon as $k > 3d/4$ (due to Heath-Brown and Browning). For the two variable case, in general we have the bound $k/d > 39/64$, due to Browning, and in the homogeneous case we have the estimate $k/d > 7/16$, also due to Browning.

A related problem is to investigate when an integral polynomial (say, of one or two variables) assumes values with very few prime factors (including multiplicity). This is a more stringent condition since many $k$-free values for example may have many distinct prime factors.

It appears to be known that in general an irreducible polynomial $f(x) \in \mathbb{Z}[x]$ assumes infinitely many values in $P_{d + 1}$, where $P_n$ denotes the set of numbers with at most $n$ prime factors (not necessarily distinct).

Is there a similar result for binary polynomials?

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I am interested in results for general polynomials, not constructing specific polynomials that hits values with few prime factors infinitely often. – Stanley Yao Xiao Dec 12 '13 at 22:14

1) The polynomial in two variables $f(x,y)=2(x^2-2y^2)$ will take the prime value $p=2$ infinitely often (Pell's equation).

2) Nontrivial results in the direction of your question are due to Gihan Marasingha:

a) Almost primes represented by binary forms. J. Lond. Math. Soc. (2) 82 (2010), no. 2, 295–316.

b) On the representation of almost primes by pairs of quadratic forms. Acta Arith. 124 (2006), no. 4, 327–355.

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The case with two variables appears easy.

There are a lot of surjective functions which satisfy your conditions.

Take: $$ f = x y $$

or Cantor bijection $$ f = \frac{1}{2} x^{2} + x y + \frac{1}{2} y^{2} + \frac{1}{2} x + \frac{3}{2} y$$

The first represents all natural numbers and the inverse map is easy.

After making the second integral it represents $2 \mathbb{N}$.

There are other integral surjections.

Or take $f=x^2+y^2$. It represents all numbers whose prime factors are of the form $4k+1$, i.e. all $P_n$.

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You could take a look at work by Gihan Marasingha. In particular ``Almost primes represented by binary forms'', Journal of the London Mathematical Society 82 (2010), 295–316). This gives a fairly general result about binary forms. It uses geometry of numbers to get better level of distributions than you would if you just had a polynomial in one variable.

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This reference has already been mentioned in another answer. Can you edit to add a newer reference or expand on how the reference answers the question? E.g. how does it use geometry of numbers? – The Masked Avenger Mar 6 '15 at 17:44

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