Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+2014\tag{1}$$

Prove or disprove: $f(x)$ is reducible over $\mathbb Q$.

See :http://www-irma.u-strasbg.fr/~bugeaud/travaux/PolyaType.pdf

Question 2:

Let $$f(x)=(x-1)^2(x-2)^2(x-3)^2\cdots(x-2013)^2+d,d\in \mathbb N^{+}.\tag{1}$$

For which $d$ is the polynomial $f(x)$ is reducible over $\mathbb Q$?

I guess that $d=4k+3,d=4k+1,k\in \mathbb N^{+}$ and $d$ is prime numbers. Is is true? Do we have other cases?

See :http://www.math.unideb.hu/~hajdul/ght20.pdf

  • $\begingroup$ Relevant: mathoverflow.net/questions/150586/… $\endgroup$ – Qiaochu Yuan Jan 6 '14 at 4:43
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    $\begingroup$ Is this for a competition? In any case, the question is not appropriate for MathOverflow. $\endgroup$ – Todd Trimble Jan 6 '14 at 4:49
  • $\begingroup$ @ToddTrimble The O.P.s (or Daniel's?) links seem to indicate that these questions are of current research interest, no matter where this question comes from. $\endgroup$ – Igor Rivin Jan 8 '14 at 0:31
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    $\begingroup$ @ToddTrimble Actually, I saw this question on MSE, where it is claimed to be a problem on the Peking U entrance exam, so the question is not genetically suitable for MO, but the links (which I am guessing are due to Daniel Litt, since they certainly are not there on MSE) indicate that the question is of interest (and, if not for the links, it would be a duplicate of the MSE question. So it is a bit of a mess, I will grant you). $\endgroup$ – Igor Rivin Jan 8 '14 at 1:34
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    $\begingroup$ @IgorRivin No, the links are due to the OP. But the extra information about MSE is helpful to me. (So they were examination problems!) I'm not completely convinced there is a close connection between the problems and the links, but I haven't thought hard about it. $\endgroup$ – Todd Trimble Jan 8 '14 at 1:38

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