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Math Help - Quadradic Progressions.

  1. #1
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    Quadradic Progressions.

    Consider the following sets.
    S_1=\{a_1n^2+b_1n+c_2|n\in \mathbb{Z}\}
    S_2=\{a_2n^2+b_2n+c_2|n\in \mathbb{Z}\}
    .....
    S_k=\{a_kn^2+b_kn+c_k|n\in \mathbb{Z}\}
    Where, a_1,a_2,...,a_k\not = 0

    Prove that,
    \bigcup_{j=1}^k S_j\not = \mathbb{Z}
    For any 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
    3k,3k+1,3k+2-they contain all the numbers.
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  2. #2
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    Consider the number of integers less than X which are represented by the form an^2 + bn + c. It is not too hard to show that this number is approximately \sqrt{X/a}. So the number of integers represented by one of k such forms is at most a quantity approximately \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.
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