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Consider a sequence of polynomials with real coefficients defined by:

$$p_0=(x^2 +1)(x^2 +2).....(x^2 +1009)$$

with subsequent polynomials defined by $$p_{k+1} (x) :=p_k (x+1) - p_k (x) $$ for $x>0$. Find the least n such that

$$p_n (1)=p_n (2)=......=p_n (5000).$$

My attempt :

Degree of the first given polynomial is 2018. And it can be seen that for $p_n (x) =2018-n$. So to have 5000 roots it must be constant function. So n=2018.

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  • $\begingroup$ Yes. That's correct. I was typing my answer/hint while your edit came. IMHO yours is the way to do this. $\endgroup$ – Jyrki Lahtonen Sep 8 '18 at 20:22
  • $\begingroup$ I am learning the use of latex. So I am slow in writing these. But thanks for clearing my doubt. :) $\endgroup$ – jayant98 Sep 8 '18 at 20:25
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Combine the following:

  • $p_0(x)$ has degree $2018$.
  • $\deg p_{k+1}=\deg p_k-1$ for all $k$.
  • For $p_n(x)-p_n(1)$ to have $5000$ zeros it must have either degree $\ge5000$ or be the constant zero.
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  • $\begingroup$ Thanks sir. Its same what I added after editing I suppose. $\endgroup$ – jayant98 Sep 8 '18 at 20:22

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