# For which constant $d$ this polynomial is reducible over $\mathbb Q$? [closed]

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 $\mathbb Q$.

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?

• Relevant: mathoverflow.net/questions/150586/… – Qiaochu Yuan Jan 6 '14 at 4:43
• Is this for a competition? In any case, the question is not appropriate for MathOverflow. – Todd Trimble Jan 6 '14 at 4:49
• @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 8 '14 at 0:31
• @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 8 '14 at 1:34
• @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 8 '14 at 1:38