2

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

Daniel Litt
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math110
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  • Relevant: http://mathoverflow.net/questions/150586/is-there-an-integer-a-such-that-fxa-is-irreducible-in-zx/150589 – Qiaochu Yuan Jan 06 '14 at 04:43
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    Is this for a competition? In any case, the question is not appropriate for MathOverflow. – Todd Trimble Jan 06 '14 at 04:49
  • @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. – Igor Rivin Jan 08 '14 at 00:31
  • @ToddTrimble So I vote to reopen. – Igor Rivin Jan 08 '14 at 00:32
  • @IgorRivin Thanks. Well, I sure wish that math110 had said something, because contest problems so often have some gimmick involving the current year (which I didn't see in the links). The form of the question is strangely specialized, don't you think? I really don't know what to think at this point (OP might have googled and found something similar-looking to his problems), but I'll consider reopening. – Todd Trimble Jan 08 '14 at 01:29
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    @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). – Igor Rivin Jan 08 '14 at 01:34
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    @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. – Todd Trimble Jan 08 '14 at 01:38
  • @ToddTrimble Only the first was an exam problem (God only knows how this was supposed to be done on a university entrance exam). The links (basically to Polya's theorem) certainly solve the problem, which is why I thought they were Daniel's. – Igor Rivin Jan 08 '14 at 01:42
  • @IgorRivin Um, hmmm... okay so do you agree that the first problem isn't really suitable for MO? The second with that silly 2013 in there makes this problem seem not of hugely compelling interest, unless it's a special case of some problem which you feel confident would be of research interest? (Maybe you should just rewrite the question in that case?) – Todd Trimble Jan 08 '14 at 01:49
  • @ToddTrimble I will try. – Igor Rivin Jan 08 '14 at 01:51
  • A suitable question has been posted by Igor: http://mathoverflow.net/questions/153902/irreducibility-of-a-class-of-polynomials – Todd Trimble Jan 08 '14 at 02:59

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