Consider the following sets.

$\displaystyle S_1=\{a_1n^2+b_1n+c_2|n\in \mathbb{Z}\}$

$\displaystyle S_2=\{a_2n^2+b_2n+c_2|n\in \mathbb{Z}\}$

.....

$\displaystyle S_k=\{a_kn^2+b_kn+c_k|n\in \mathbb{Z}\}$

Where, $\displaystyle a_1,a_2,...,a_k\not = 0$

Prove that,

$\displaystyle \bigcup_{j=1}^k S_j\not = \mathbb{Z}$

For any $\displaystyle k\geq 1$

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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.