
Quadradic Progressions.
Consider the following sets.
$\displaystyle S_1=\{a_1n^2+b_1n+c_2n\in \mathbb{Z}\}$
$\displaystyle S_2=\{a_2n^2+b_2n+c_2n\in \mathbb{Z}\}$
.....
$\displaystyle S_k=\{a_kn^2+b_kn+c_kn\in \mathbb{Z}\}$
Where, $\displaystyle a_1,a_2,...,a_k\not = 0$
Prove that,
$\displaystyle \bigcup_{j=1}^k S_j\not = \mathbb{Z}$
For any $\displaystyle k\geq 1$

What I am trying to show, informally, that a set of integers cannot be placed in classes of quadradic progressions.
For example, with linear progressions it is always possible. Consider
$\displaystyle 3k,3k+1,3k+2$they contain all the numbers.

Consider the number of integers less than X which are represented by the form $\displaystyle an^2 + bn + c$. It is not too hard to show that this number is approximately $\displaystyle \sqrt{X/a}$. So the number of integers represented by one of k such forms is at most a quantity approximately $\displaystyle \sqrt{X} \sum_{i=1}^k \frac1{\sqrt{a_i}}$ and for large enough X this must be less than X. Hence some numbers are not so represented.